# Mathematics Placement

Size: px
Start display at page:

Download "Mathematics Placement"

Transcription

1

2 Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry. As you answer questions correctly, you will move into more difficult levels of math. Similarly, if you answer questions incorrectly, the computerized test will begin to ask questions from a lower level of math. Multiple-choice items in each of the five mathematics placement areas test the following: basic skills performing a sequence of basic operations application applying sequences of basic operations to novel settings or in complex ways analysis demonstrating conceptual understanding of principles and relationships in mathematical operations Students are permitted to use approved calculators when completing the COMPASS mathematics placement or diagnostic tests. An online calculator is available for those students who wish to access it via Microsoft Windows. Because this is an adaptive test, you may change your answer while you are still on a problem, but once you go on to another problem, you may not go back to a question.

3 Math Diagnostics Test Depending on the results of you Math Placement Test, you may be required to take the Math Diagnostic Test. This math test evaluates students' skill levels in 15 subareas in Pre-Algebra and Algebra: Pre-Algebra Integers Decimals Exponents, square roots, and scientific notation Fractions Percentages Averages (means, medians, and modes) Algebra Substituting values Setting up equations Factoring polynomials Exponents and radicals Basic operations/polynomials Linear equations/one variable Linear equations/two variables Rational expressions

4 Mathematics Placement Sample Questions (PreAlgebra) Following are 16 sample Algebra Placement Test Questions taken from the ACT COMPASS website. First you will see the question, then the following slide will have the answer. If you need some additional refreshers, the remainder of the slides cover the content from the Algebra section.

5 Algebra Placement Test Sample Questions

6 Algebra Placement Test Sample Questions This is an example of Substituting Values into Algebraic Expressions. The correct answer is A (-4). You would need to be familiar with Evaluating Expressions to correctly answer this question (see Evaluating Expressions slides for additional information on this topic). To solve: Step 1: Substitute value into expression Step 2: Solve ( 3)

7 Algebra Placement Test Sample Questions 2. Doctors use the term maximum heart rate (MHR) when referring to the quantity found by starting with 220 beats per minute and subtracting 1 beat per minute for each year of a person s age. Doctors recommend exercising 3 or 4 times each week for at least20 minutes with your heart rate increased from its resting heart rate (RHR) to its training heart rate (THR), where THR = RHR +.65(MHR RHR) Which of the following is closest to the THR of a 43-year-old person whose RHR is 54 beats per minute? A. 197 B. 169 C. 162 D. 134 E. 80

8 Algebra Placement Test Sample Questions 2. Doctors use the term maximum heart rate (MHR) when referring to the quantity found by starting with 220 beats per minute and subtracting 1 beat per minute for each year of a person s age. Doctors recommend exercising 3 or 4 times each week for at least 20 minutes with your heart rate increased from its resting heart rate (RHR) to its training heart rate (THR), where THR = RHR +.65(MHR RHR) Which of the following is closest to the THR of a 43-year-old person whose RHR is 54 beats per minute? A. 197 B. 169 C. 162 D. 134 E. 80 This is an example of Substituting Values into Algebraic Expressions. The correct answer is D (134). You would need to be familiar with Evaluating Expressions to correctly answer this question (see Evaluating Expressions slides for additional information on this topic). To solve: THR = [(220-1(43)) 54] = (123) = =

9 Algebra Placement Test Sample Questions 3. When getting into shape by exercising, the subject s maximum recommended number of heartbeats per minute (h) can be determined by subtracting the subject s age (a) from 220 and then taking 75% of that value. This relation is expressed by which of the following formulas? A. h =.75(220 a) B. h =.75(220) a C. h = a D..75h = 220 a E. 220 =.75(h a)

10 Algebra Placement Test Sample Questions 3. When getting into shape by exercising, the subject s maximum recommended number of heartbeats per minute (h) can be determined by subtracting the subject s age (a) from 220 and then taking 75% of that value. This relation is expressed by which of the following formulas? A. h =.75(220 a) B. h =.75(220) a C. h = a D..75h = 220 a E. 220 =.75(h a) This is an example of Setting Up Equations for Given Situations. The correct answer is A (h =.75(220 a)). You would need to be familiar with Writing Equations to correctly answer this question (see Writing Equations slides for additional information on this topic). To solve: Remember to change the percent to a decimal: 75% =.75. Other key words are subtract from (-) and of (x).

11 Algebra Placement Test Sample Questions 4. An airplane flew for 8 hours at an airspeed of x miles per hour (mph), and for 7 more hours at 325 mph. If the average airspeed for the entire flight was 350 mph, which of the following equations could be used to find x? A. x = 2(350) B. x + 7(325) = 15(350) C. 8x 7(325) = 350 D. 8x + 7(325) = 2(350) E 8x + 7(325) = 15(350)

12 Algebra Placement Test Sample Questions 4. An airplane flew for 8 hours at an airspeed of x miles per hour (mph), and for 7 more hours at 325 mph. If the average airspeed for the entire flight (which was 15 hours total) was 350 mph, which of the following equations could be used to find x? A. x = 2(350) B. x + 7(325) = 15(350) C. 8x 7(325) = 350 D. 8x + 7(325) = 2(350) E. 8x + 7(325) = 15(350) This is an example of Setting Up Equations for Given Situations. The correct answer is E (8x + 7(325) = 15(350) ). You would need to be familiar with Writing Equations to correctly answer this question (see Writing Equations slides for additional information on this topic). To solve: See key words in the question.

13 Algebra Placement Test Sample Questions 5. Which of the following is equivalent to 3a + 4b ( 6a 3b)? A. 16ab B. 3a + b C. 3a + 7b D. 9a + b E. 9a + 7b

14 Algebra Placement Test Sample Questions 5. Which of the following is equivalent to 3a + 4b ( 6a 3b)? A. 16ab B. 3a + b C. 3a + 7b D. 9a + b E. 9a + 7b This is an example of Basic Operations with Polynomials. The correct answer is E (9a + 7b). You would need to be familiar with Combining Like Terms to correctly answer this question (see Basic Operations with Polynomials slides for additional information on this topic). To solve: Distribute the negative through the parenthesis and then combine like terms: 3a + 4b ( 6a 3b) = 3a + 4b + 6a + 3b = (3a + 6a) + (4b + 3b) = 9a + 7b

15 Algebra Placement Test Sample Questions

16 Algebra Placement Test Sample Questions This is an example of Basic Operations with Polynomials. The correct answer is A (3a 2 b ab 2 + 3a 2 b 2 ). You would need to be familiar with Adding Polynomials and Combining Like Terms to correctly answer this question (see Basic Operations with Polynomials slides for additional information on this topic). To solve: Sum means add the two polynomials, then combine like terms: (3a 2 b + 2a 2 b 2 )+ (-ab 2 +a 2 b 2 ) = 3a 2 b - ab 2 + (2a 2 b 2 + a 2 b 2 ) = 3a 2 b ab 2 + 3a 2 b 2

17 Algebra Placement Test Sample Questions 7. Which of the following is a factor of the polynomial x 2 x 20? A. x 5 B. x 4 C. x + 2 D. x + 5 E. x + 10

18 Algebra Placement Test Sample Questions 7. Which of the following is a factor of the polynomial x 2 x 20? A. x 5 B. x 4 C. x + 2 D. x + 5 E. x + 10 This is an example of Factoring Polynomials. The correct answer is A (x-5). You would need to be familiar with Factoring Polynomials to correctly answer this question (see Factoring Polynomials slides for additional information on this topic). To solve: Factor the quadratic equation: x 2 x 20 = (x 5)(x + 4). Check using FOIL.

19 Algebra Placement Test Sample Questions 8. Which of the following is a factor of x 2 5x 6? A. (x + 2) B. (x 6) C. (x 3) D. (x 2) E. (x 1)

20 Algebra Placement Test Sample Questions 8. Which of the following is a factor of x 2 5x 6? A. (x + 2) B. (x 6) C. (x 3) D. (x 2) E. (x 1) This is an example of Factoring Polynomials. The correct answer is B (x - 6). You would need to be familiar with Factoring Polynomials to correctly answer this question (see Factoring Polynomials slides for additional information on this topic). To solve: Factor the quadratic equation: x 2 5x 6 = (x 6)(x + 1). Check using FOIL.

21 Algebra Placement Test Sample Questions

22 Algebra Placement Test Sample Questions This is an example of Linear Equations in One Variable. The correct answer is E (-½). You would need to be familiar with Solving Linear Equations to correctly answer this question (see Solving Linear Equations slides for additional information on this topic). To solve: Solve for x: 2( x x x 5 5)

23 Algebra Placement Test Sample Questions

24 Algebra Placement Test Sample Questions This is an example of Linear Equations in One Variable. The correct answer is C (-1). You would need to be familiar with Solving Linear Equations to correctly answer this question (see Solving Linear Equations slides for additional information on this topic). To solve: Solve for x: x x x x

25 Algebra Placement Test Sample Questions

26 Algebra Placement Test Sample Questions This is an example of Exponents. The correct answer is B. You would need to be familiar with Rules for Exponents to correctly answer this question (see Rules for Exponents slides for additional information on this topic). To solve: Apply rules for exponents: r tz 3 2 4rt z (16 4)( r 3 r)( t t 3 )( z 5 z 2 ) 4r 2 t 2 z 3 4r t 2 2 z 3

27 Algebra Placement Test Sample Questions

28 Algebra Placement Test Sample Questions x 3 x y x 3 x y 3 x y 3 x y 9x y This is an example of Exponents. The correct answer is C. You would need to be familiar with Rules for Exponents to correctly answer this question (see Rules for Exponents slides for additional information on this topic). To solve: First, you want to get rid of the radical signs in the denominator. To do that you would multiply the numerator and the denominator by the expression. 3x xy

29 Algebra Placement Test Sample Questions

30 Algebra Placement Test Sample Questions This is an example of Rational Expressions. The correct answer is B (x + 8). You would need to be familiar with Rational Expressions to correctly answer this question (see Rational Expressions slides for additional information on this topic). To solve: x 12x 32 x 4 ( x 4)( x 8) x 4 x 8 2 Step 1: Factor the numerator. Step 2: Cancel common terms.

31 Algebra Placement Test Sample Questions

32 Algebra Placement Test Sample Questions This is an example of Rational Expressions. The correct answer is E (-x 3). You would need to be familiar with Rational Expressions to correctly answer this question (see Rational Expressions slides for additional information on this topic). To solve: 9 x 2 x 3 ( x ( x ( x x 3)( x x 3 3) 2 x 9) 3 3) 3 Step 1: Factor the numerator. Step 2: Cancel common terms. Step 3: Distribute the neg. back through

33 Algebra Placement Test Sample Questions

34 Algebra Placement Test Sample Questions This is an example of Linear Equations in Two Variables. The correct answer is D ( 2/3). You would need to be familiar with Linear Equations to correctly answer this question (see Linear Equations slides for additional information on this topic). Use the slope intercept form: y=mx+b where m=slope and b=y-intercept 2x+3y+6 = 0 (standard form) Solve for y: 3y = -2x 6 y = (-2/3)x 2 Therefore, the slope is 2/3

35 Algebra Placement Test Sample Questions

36 Algebra Placement Test Sample Questions This is an example of Linear Equations in Two Variables. The correct answer is E ((8, 1)). You would need to be familiar with Linear Equations to correctly answer this question (see Linear Equations slides for additional information on this topic). Graph the point and the line. A perpendicular (cross at 90 degree angle) bisector (cuts the line segment into two equal length segments). Also, for future reference, parallel lines have the same slope, and perpendicular lines have negative reciprocal slope.

37 Algebra Review The following slides review the concepts found on the COMPASS Algebra Placement Test.

38 Substituting Values 1. Evaluating Variable Expressions

39 Evaluating Variable Expressions Section 2.1 Variable - a letter that is used to stand for a quantity that is unknown or may change Variable Expression - an expression that contains variables Terms - addends of a variable expression Ex: the expression 4x 2 + 3x - 2y + 5 has four terms that include three variable terms (4x 2, 3x, 2y) and one constant term (5) Coefficient is the number part of the variable term. In the term 4x 2 the 4 is the coefficient.

40 Note x = 1x -x = -1x xy = x times y

41 Evaluating the Variable Expression Evaluating the Variable Expression - replace the variable with numbers and then simplify EX: xy - 3 where x = 5, y = 4 replace 5(4) - 3 = 20-3 = 17 Don t forget Order of Operations - PEMDAS Parenthesis, Exponents, Mult/Div, Add/Subtract

42 Practice: Evaluate Ex1: ab 2 a when a = 2 and b = -2 Ex2: a 2 b 2 when a = 3 and b = -4 a - b Ex3: a 2 + b 2 when a = 5 and b = -3 a + b

43 Practice: Answers Ex1: ab 2 a when a = 2 and b = -2 2( 2) 2 2 2(4) Ex2: a 2 b 2 when a = 3 and b = -4 a - b ( ( 4) 4) Ex3: a 2 + b 2 when a = 5 and b = -3 a + b ( ( 3) 3) ( 9 3)

44 You Try: Evaluate Ex1: 3a + 2b when a = 2 and b = 3 Ex2: 2a (c + a) 2 when a = 2 and c = -4 Ex3: a 2 5a 6 when a = -2 For Ex 4-7 use: a = -2, b = 4, c = -1, d = 3 Ex4: (b + c) 2 + (a + d) 2 Ex5: b 2a bc 2 -d Ex6: b 2 - a ad + 3c Ex7: 1/3(d 2 ) 3/8(b 2 )

45 You Try: Answers Ex1: 3a + 2b when a = 2 and b = 3 3(2) 2(3) 12 Ex2: 2a (c + a) 2 when a = 2 and c = -4 2(2) ( 4 2) Ex3: a 2 5a 6 when a = -2 ( 2) 2 5( 2 2) 6 6 2(2) 6 2(2) For Ex 4-7 use: a = -2, b = 4, c = -1, d = 3 Ex4: (b + c) 2 + (a + d) (4 1) ( 2 3) 4 ( 2) Ex5: b 2a bc 2 -d 4 4( 2( 1) 2 2) 3 4 4(1) Ex6: b 2 - a ad + 3c 4 2 2(3) ( 2) 3( 1) 16 6 ( ( 2) 3) Ex7: 1/3(d 2 ) 3/8(b 2 ) (3 3 ) 3 (4 8 ) 1 3 (9) 3 8 (16) 3 6 3

46 Setting Up Equations 1. Phrases that mean Addition 2. Phrases that mean Subtraction 3. Phrases that mean Multiplication 4. Phrases that mean Division 5. Phrases that mean Exponents 6. Translating Words into Expressions

47 Phrases that mean Addition Section 2.6 added to 6 added to y y + 6 more than 8 more than x x + 8 the sum of the sum of x and z x + z increased by t increased by 9 t + 9 the total of the total of 5 and y 5 + y plus b plus 17 b + 17 Other words that mean addition: older, greater, additional, longer

48 Phrases that mean Subtraction Section 2.6 minus x minus 2 x - 2 less than 7 less than t t - 7 less 7 less t 7 - t subtracted from 5 subtracted from d d - 5 decreased by m decreased by 3 m - 3 the difference between the difference between y and 4 y - 4 Additional words that mean subtraction: younger, fewer, shorter Note: The book uses less than and difference frequently. Make sure you know the difference between them!

49 Phrases that mean Multiplication Section 2.6 times 10 times t 10t of one half of x (1/2)x the product of the product of y and z yz multiplied by y multiplied by 11 11y twice twice n 2n

50 Phrases that mean Division Section 2.6 divided by x divided by 12 x/12 the quotient of the quotient of y and z y/z the ratio of the ratio of t to 9 t/9 Additional words that mean division: divided among, split between

51 Phrases that mean Exponents (Powers) Section 2.6 the square of the square of x x 2 the cube of the cube of a a 3

52 Translating Phrases into Expressions Section 2.6 Note: Remember that addition and multiplication are commutative, so order doesn t matter. However, subtraction and division are not commutative, so order DOES matter.

53 S e c t i o n 2. 6 Word Phrase Variable (can be any letter) 4 less than a number a a 4 A number decreased by 9 b b 9 50 minus a number z 50 z A number more than 5 x 5 + x A number increased by 7 y y + 7 the sum of 6 and a number c 6 + c 3 plus a number k 3 + k The difference of a number and 8 n n 8 7 times a number m 7m The product of 12 and a number t 12t 3/4 of a number k ¾k A number times 8 d d x 8 A number divided by 2 y y/2 6 divided by a number s 6/s Expression

54 Translate into a variable expression The total of five times b and c 5b + c Identify words that indicate mathematical operations. Use the operations to write the variable expression.

55 Translate into a variable expression The quotient of eight less than n and fourteen n 8 14 Identify words that indicate mathematical operations. Use the operations to write the variable expression.

56 Translate into a variable expression Thirteen more than the sum of seven and the square of x Identify words that indicate mathematical operations. (7 + x 2 ) + 13 Use the operations to write the variable expression.

57 Your turn! Translate into a variable expression eighteen less than the cube of x y decreased by the sum of z and nine the difference between the square of q and the sum of r and t

58 Your turn! (answers) Translate into a variable expression eighteen less than the cube of x x 3-18 y decreased by the sum of z and nine y (z + 9) the difference between the square of q and the sum of r and t q 2 (r + t)

59 More Examples Translate a number multiplied by the total of six and the cube of the number into a variable expression. Break into parts The unknown number: n The cube of a number: n 3 The total of six and the cube of the number: 6 + n 3 Put it together n(6 + n 3 )

60 More Examples Translate the quotient of twice a number and the difference between the number and twenty into a variable expression. Break into parts The unknown number: n Twice the number: 2n The difference between the number and twenty: n - 20 Put it together 2n n - 20

61 Your turn! Translate a number added to the product of five and the square of the number into a variable expression. Translate the product of three and the sum of seven and twice a number into a variable expression.

62 Your turn! (answers) Translate a number added to the product of five and the square of the number into a variable expression. 5n 2 + n Translate the product of three and the sum of seven and twice a number into a variable expression. 3(7 + 2n)

63 More Examples After translating a verbal expression into a variable expression, simplify the variable expression by using the Properties of Real Numbers. Translate and simplify the total of four times an unknown number and twice the difference between the number and eight. Break into parts The unknown number: n Four times the number: 4n Twice the difference between the number and eight: 2(n 8) Put it together: 4n + 2(n - 8) Simplify: 4n + 2n 16 = 6n - 16

64 Your turn! Translate and simplify a number minus the difference between twice the number and seventeen. n (2n 17) n 2n n + 17

65 Factoring Polynomials 1. Greatest Common Factor and Factoring by Grouping 2. Factoring Trinomials: x 2 + bx + c 3. More on Factoring Trinomials: ax 2 + bx + c 4. Factoring Special Products: Difference of Two Squares and Perfect Square Trinomials

66 Greatest Common Factor Factoring is the process of breaking a product into smaller parts called factors. Recall that the product is the answer to a multiplication problem and the factors are the terms being multiplied. Examples: For 2 5 = 10, the factors are 2 and 5. For (x - 2)(x + 4) = 0, the factors are x - 2 and x + 4. Common Factor: If all terms in the equation have a common factor, you can factor it out and have a simpler equation to solve. Try to find the greatest common factor (GCF) to factor out.

67 Greatest Common Factor Find the GCF for each of the following: Ex 1: 16, 40, = 2x2x2x2 = = 2x2x2x5 = = 2x2x2x7 = so the GCF = 2 3 = 8 Ex 2: 45x 2 y 2 z 2 and 75xy 2 z 3 45x 2 y 2 z 2 = x 2 y 2 z 2 75xy 2 z 3 = x y 2 z 3 so the GCF = 3 5 x y 2 z 2 = 15xy 2 z 2

68 Greatest Common Factor Example: Factor the expression 8x The common factors are 1, 2, and 4 because they all divide evenly into both 8x and 12. The GCF is 4 and can be factored out. Divide 8x by 4, which is 2x. Divide 12 by 4, which is 3. The factored equation is 4(2x - 3). TIP: Factoring out a common factor is the opposite of applying the distributive property.

69 Greatest Common Factor Factor each of the polynomials by finding the greatest common monomial factor: Ex 1: 14x + 21 Ex 2: Ex 3: Ex 4: -3x 2 + 6x -6ax + 9ay 16x 4 y 14x 2 y Ex 5: 34x 4 y 6 51x 3 y x 5 y 4

70 Greatest Common Factor Factor each of the polynomials by finding the greatest common monomial factor: Ex 1: 14x + 21 = 7(2x + 3) Ex 2: -3x 2 + 6x = -3x(x 2) Ex 3: -6ax + 9ay = -3a(2x - 3y) Ex 4: 16x 4 y 14x 2 y = 2x 2 y(8x 2 7) Ex 5: 34x 4 y 6 51x 3 y x 5 y 4 = 17x 3 y 4 (2xy 2 3y + x 2 ) or 17x 3 y 4 (x 2 3y + 2xy 2 )

71 Factor by Grouping Consider the expression y(x + 2) + (x + 2) as the sum of two terms, y(x + 2) and 1(x +2). Each of these terms have a common binomial factor of (x + 2). Factoring out this common binomial factor by using the distributive property gives y(x + 2) + (x + 2) = (y + 1)(x + 2)

72 Factor by Grouping Section 7.1 Factor each expression by factoring out the common binomial factor: Ex 1: 7y 2 (y + 3) + 2(y + 3) Ex 2: 9a(x + 1) (x + 1) Ex 3: 2x 2 (x + 5) + (x + 5) Ex 4: 10y(2y + 3) 7(2y + 3) Ex 5: a(x 2) b(x 2)

73 Factor by Grouping Factor each expression by factoring out the common binomial factor: Ex 1: 7y 2 (y + 3) + 2(y + 3) = (7y 2 + 3)(y + 3) Ex 2: 9a(x + 1) (x + 1) = (9a 1)(x + 1) Ex 3: 2x 2 (x + 5) + (x + 5) = (2x 2 + 1)(x + 5) Ex 4: 10y(2y + 3) 7(2y + 3) = (10y 7)(2y + 3) Ex 5: a(x 2) b(x 2) = (a b)(x 2)

74 Factor by Grouping Factor each polynomial by grouping. If the polynomial cannot be factored, write not factorable: Ex 1: bx + b + cx + c Ex 2: x 3 + 3x 2 + 6x + 18 Ex 3: 24y 3yz + 2xz 16x Ex 4: 10xy + x y 1 Ex 5: x x 2 y + 5y

75 Factor by Grouping Factor each polynomial by grouping. If the polynomial cannot be factored, write not factorable: Ex 1: bx + b + cx + c = b(x + 1) + c(x + 1) = (b + c)(x + 1) Ex 2: x 3 + 3x 2 + 6x + 18 = x 2 (x + 3) + 6(x + 3) = (x 2 + 6)(x + 3) Ex 3: 24y 3yz + 2xz 16x = 3yz + 24y + 2xz 16x = = -3y(z 8) + 2x(z 8) = (2x 3y)(z 8) Ex 4: 10xy + x y 1 = 10x(y + 1) (y + 1) = (10x 1)(y + 1) Ex 5: x x 2 y + 5y = (x 2 5) + y(x 2 + 5) not factorable

76 Factor Trinomials: x 2 + bx + c Quadratic equations are written in this form y = ax 2 + bx + c where ax 2 is the quadratic term (to the 2 nd power) bx is the linear term (to the 1 st power) c is the constant term Notice that the variable terms are listed in decreasing order.

77 Factor Trinomials: x 2 + bx + c Example: Multiply (x - 1)(x + 4) FOIL (First, Outside, Inside, Last) 1x 2 + 4x - x - 4 x 2 + 3x - 4 Combine like terms. Result is a trinomial Notice that the result of multiplying binomials is a trinomial. Factoring a trinomial requires finding the two binomials that were originally multiplied together to give the trinomial. The process of factoring involves working backwards from the terms in the trinomial.

78 Factor Trinomials: x 2 + bx + c To factor a trinomial with leading coefficient 1, find the two factors of the constant term whose sum is the coefficient of the middle term. x 2 + (a + b)x + ab Sum of constants a and b Product of constants a and b

79 Factor Trinomials: x 2 + bx + c Ex: Factor: x 2 + 7x + 12 To factor x 2 + 7x + 12, find the two factors of 12 whose sum is 7. x 2 + (3 + 4)x + 3(4) Sum of constants a and b Product of constants a and b Therefore, x 2 + 7x + 12 = (x + 3)(x + 4)

80 Factor Trinomials: x 2 + bx + c Ex: Factor: x 2-8x + 16 To factor x 2-8x + 16, find the two factors of 16 whose sum is -8. x 2 + ( )x + -4(-4) Sum of constants a and b Product of constants a and b Therefore, x 2-8x + 16 = (x - 4)(x - 4)

81 Factor Trinomials: x 2 + bx + c Ex: Factor: x 2 + 4x 21 To factor x 2 + 4x 21, find the two factors of -21 whose sum is +4. x 2 + (7 + -3)x + 7(-3) Sum of constants a and b Product of constants a and b Therefore, x 2 + 4x 21 = (x +7)(x 3)

82 Factor Trinomials: x 2 + bx + c We have seen from the previous 3 examples that a rule can be formed: To factor x 2 + bx + c, if possible, find an integer pair of factors of c whose sum is b. 1. If c is positive, then both factors must have the same sign. a. Both will be positive if b is positive b. Both will be negative if b is negative 2. If c is negative, then one factor must be positive and the other negative.

83 Factor Trinomials: x 2 + bx + c Factor the trinomials, if the trinomial cannot be factored, write not factorable: Ex 1: x 2 x 12 Ex 2: y 2 3y + 2 Ex 3: y y + 35 Ex 4: a 2 + a + 2

84 Factor Trinomials: x 2 + bx + c Section 7.2 Factor the trinomials, if the trinomial cannot be factored, write not factorable: Ex 1: x 2 x 12 = (x + 3)(x - 4) Ex 2: y 2 3y + 2 = (y 1)(y 2) Ex 3: y y + 35 = (y + 5)(y + 7) Ex 4: a 2 + a + 2 is not factorable

85 Factoring Trinomials: ax 2 + bx + c A polynomial is completely factored if none of its factors can be factored. To factor ax 2 + bx + c, where a 1, look for a common monomial factor. If there is a common monomial factor, factor out this monomial factor and factor the remaining trinomial, if possible. We will discuss other options in section 7.3.

86 Factoring Trinomials: ax 2 + bx + c Completely factor the polynomials: Ex 1: 5x 2 5x 60 Ex 2: 7y 3 70y y Ex 3: 3x 2 18x + 30 Ex 4: a a a 2 Ex 5: 20a ab + 20b 2

87 Factoring Trinomials: ax 2 + bx + c Completely factor the polynomials: Ex 1: 5x 2 5x 60 = 5(x 2 x 12) = 5(x 4)(x + 3) Ex 2: 7y 3 70y y = 7y(y 2 10y + 24) = 7y(y 4)(y 6) Ex 3: 3x 2 18x + 30 = 3(x 2 6x + 10) Ex 4: a a a 2 = a 2 (a a + 81) = a 2 (a + 3)(a + 27) Ex 5: 20a ab + 20b 2 = 20(a 2 +2ab + b 2 ) = 20(a + b)(a + b)

88 Factoring Trinomials: ax 2 + bx + c Factoring using the ac-method: 1. Multiply a by c 2. Find two integers whose product is ac and whose sum b 3. Rewrite the middle term bx using the two numbers found in Step 2 as coefficients 4. Factor by grouping the first two terms and the last two terms 5. Factor out the common binomial factor to find two binomial factors of the trinomial ax 2 + bx + c

89 Factoring Trinomials: ax 2 + bx + c Factoring 4x 2-5x - 6 using the ac-method: 1. Multiply a by c: 4(-6) = Find two integers whose product is ac and whose sum b: 3(-8) and Rewrite the middle term bx using the two numbers found in Step 2 as coefficients: 4x 2-8x + 3x Factor by grouping the first two terms and the last two terms: 4x(x - 2) + 3(x 2) 5. Factor out the common binomial factor to find two binomial factors of the trinomial ax 2 + bx + c: (4x + 3)(x 2)

90 Factoring Trinomials: ax 2 + bx + c Factoring using the Trial by Error Method: 1. The key is using the FOIL method (First, Outside, Inside, Last)

91 Factor: Factoring Trinomials: ax 2 + bx + c Ex 1: 2x 2 3x 2 Ex 2: 12y 2 15y + 3 Ex 3: -5y y - 60

92 Factoring Trinomials: ax 2 + bx + c Factor: Ex 1: 2x 2 3x 2 = 2x 2 4x + 1x 2 = 2x(x 2) + (x 2) = (2x + 1)(x 2) Ex 2: 12y 2 15y + 3 = 3(4y 2 5y + 1) = 3[4y 2 4y 1y + 1] = 3[4y(y 1) (y 1)] = 3(4y 1)(y 1) Ex 3: -5y y 60 = -5(y 2-8y + 12) = -5(y - 2)(y - 6)

93 Difference of Squares When factoring the difference of two squares: x 2 a 2 = (x + a)(x a) Ex 1: x 2 25 = (x + 5)(x 5) Ex 2: 36 y 2 = (6 + y)(6 y) Ex 3: a = (a )(a 3 10)

94 Perfect Square Trinomials When factoring perfect square trinomials: x 2 + 2ax + a 2 = (x + a) 2 Ex 1: x x = (x + 10)(x + 10) = (x + 10) 2 Ex 2: y + y 2 = (7 + y) 2 Ex 3: 4a a = (2a 3 + 5) 2

95 Perfect Square Trinomials Section 7.4 When factoring perfect square trinomials: x 2 2ax + a 2 = (x - a) 2 Ex 1: x 2 20x = (x 10)(x 10) = (x 10) 2 Ex 2: 49 14y + y 2 = (7 y) 2 Ex 3: 4a 6 20a = (2a 3 5) 2

96 Factoring Trinomials: ax 2 + bx + c Procedures to follow when factoring Polynomials: 1. Factor out any common monomial factor. 2. Check the number of terms: a. Two terms: 1) Difference of two squares? factorable 2) Sum of two squares? not factorable b. Three terms: 1) Perfect square trinomial? 2) Use trial-and-error method? 3) Use the ac-method? c. Four terms: 1) Group terms with a common factor 3. Check the possibility of factoring any of the factors

97 Factoring Trinomials: ax 2 + bx + c We can add to a binominal the constant term that makes it a perfect-square trinomial ; this process is called Completing the Square. In this case, (½ of our coefficient of our linear term) 2 = constant term. To complete the squares we want: y y + = ( ) 2 to be in the form x 2 + 2ax + a 2 = (x + a) 2 so ½ (b) = ½ (20) = 10. Therefore, we would have y 2 +20y = (y + 10) 2

98 Factoring Trinomials: ax 2 + bx + c Complete the squares for the following: Ex 1: x 2 6x + = ( ) 2 Ex 2: x 2 4x + = ( ) 2 Ex 3: x = ( ) 2 Ex 4: x = ( ) 2

99 Factoring Trinomials: ax 2 + bx + c Complete the squares for the following: Ex 1: x 2 6x + 9 = (x - 3) 2 Ex 2: x 2 4x + 4 = (x - 2) 2 Ex 3: x 2 + 8x + 16 = (x + 4) 2 Ex 4: x 2-18x + 81 = (x - 9) 2

100 Exponents and Radicals 1. Properties of Exponents 2. Simplifying Radicals 3. Properties of Radicals

101 Exponents Exponent: In the expression x n (read: x to the nth power), where x is the base and n is the exponent. x n the base (x) is multiplied by itself exponent (n) number of times Ex: 5 2 = 5 x 5 = = 2 x 2 x 2 = 8

102 Be careful (-3) 2 does not equal -3 2 it means (-3)(-3) = means (3 x 3) = -9

103 Exponents When dealing with particular operations, there are rules for simplifying or evaluating an exponential expression. Exponent of 1: a 1 = a Exponent of 0: a 0 = 1, when a 0 Negative Exponents: a -n = 1 a n = 1/a n, where a 0 1 to a Power: 1 n = 1 To add and subtract exponential expressions, like bases with like exponents are required. Example: LIKE bases with LIKE exponents can be simplified. a 2 + a 2 = 2a 2 5b 2-3b 2 = 2b 2

104 Properties of Exponents Rule: a m a n = a m+n To multiply two or more exponential expressions that have the same base, add the exponents. Example: 5 3 x 5 2 = 5 (3+2) = 5 5 Rule: a m a n = a m /a n = a m-n where a 0 Rule: (a m ) n = a m(n) Rule: (ab) m = a m b m To divide two or more exponential expressions that have the same base, subtract the exponents. Example: = 5 (3-2) = 5 1 = 5 To raise a power to a power, multiply the exponents. Example: (a 6 ) 5 = a 6x5 = a 30 To raise a product to a power, raise each factor to that power. Example: (ab) 5 = a 5 b 5 Rule: m m a a m b b where b 0 To raise a fraction to a power, raise both the numerator and the denominator to that power. Example:

105 Simplifying Radicals An expression with radicals is in simplest from if the following are true. 1. No radicands (expressions under radical signs) have perfect square factors other than 1. Example: No radicands contain fractions. Example: No radicals appear in the denominator of a fraction. Example: Note: To simplify this expression, multiply the numerator and denominator by 2. This is algebraically justified because it is equivalent to multiplying the original fraction by 1.

106 Properties of Radicals Rule: ab a b The square root of a product equals the product of the square roots of the factors. Example: Rule: a b a b The square root of a quotient equals the quotient of the square roots of the numerator and denominator. Example:

107 Examples: = or = 8 3 = = x 3 x 2 = x 3+2 = x y y 4 = 3y 1+4 = 3y 5 4. x -3 = 1/ x x 5 /2x 2 = -5x 5-2 = -5x a b c 5 9a b x x 2 3 y 4a 5 ( 5) xy b 0 ( 3) c 2 4a 10 b 3 c 2 4a c 10 2 b 3 y y y y

108 Basic Operations of Polynomials 1. Addition with Polynomials 2. Subtraction with Polynomials 3. Multiplication with Polynomials

109 Polynomials Vocabulary A monomial is an algebraic expression that contains only one term (i.e. -2x 3 and 4a 5 ). A binomial is an algebraic expression that contains exactly two unlike terms (i.e. 3x + 5 and a 2 3). A trinomial is an algebraic expression that contains exactly three unlike terms (a 2 + 6a 7 and x 3 8x x). A polynomial is an algebraic expression that contains one or more unlike terms. The degree of the polynomial is the largest of the degrees of its terms. The coefficient of the term of the largest degree is called the leading coefficient. To simplify polynomials means to add or subtract any like terms and when possible write it in order of descending exponents.

110 Adding Polynomials Simplifying Polynomials The sum of two or more polynomials is found by combining like terms. Remember that like terms are constants or terms that contain the same variable raised to the same powers. Ex: Find the sum: (10x x - 91) + (12x x - 95) To easily see the like terms, rewrite the polynomials vertically so that the like terms line up. Then add the coefficients of the variables. 10x x x x x x (This is written in simplest form because it contains no like terms.)

111 Adding Polynomials Simplifying Polynomials Ex: Find the sum: (7x 3 + 5x 2 + x 6) + (-3x 2 +11) + (-3x 3 x 2 5x +2) 7x 3 + 5x 2 + x 6-3x x 3 x 2 5x +2 4x 3 + x 2 4x +7

112 Subtracting Polynomials Simplifying Polynomials Remember that a negative sign written in front of a parentheses, means to change the sign of every term within the parenthesis. Ex: -(12x x - 95) = -12x 2-29x + 95 The difference of two polynomials is found by changing the sign of each term of the second polynomial and then combining like terms. Ex: Find the difference: (10x x - 91) - (12x x - 95) To easily see the like terms, rewrite the polynomials vertically so that the like terms line up. Then add the coefficients of the variables. 10x x x 2-29x x 2-12x + 4 (This is written in simplest form because it contains no like terms.)

113 Adding & Subtracting Polynomials Simplifying Polynomials Ex 1: Simplify and write in descending order: 4x + 2(x 3) (3x + 7) 4x + 2x 6 3x 7 Distribute 3x 13 Combine like terms Ex 2: Simplify and write in descending order: -[6x 2 3(4 + 2x) + 9] (x 2 + 5) Distribute inner most parenthesis -[6x x + 9] x 2 5-6x x - 9 x 2 5 Combine like terms -7x 2 + 6x 2 Distribute negative through bracket

114 Multiplying Polynomials Multiply Polynomials Alg.: D To multiply a polynomial by a monomial, distribute each term to the others by multiplication. Example: Multiply 2z(z ). 2z(z ) Follow the colors. 2z 3-10z + 16z 2z 3 + 6z Combine like terms

115 Multiplying Polynomials Multiply Polynomials Multiplying Binomials: Example: (x - 1)(x + 4) FOIL (first, outside, inside, last) 1x 2 + 4x - x - 4 x 2 + 3x - 4 Combine like terms Notice that the result of multiplying binomials is a trinomial.

116 Multiplying Polynomials Multiply Polynomials Alg.: D Multiply and simplify: 8x 2 + 3x 2-2x x 2 +21x 14 Multiply each term of the first trinomial by +7-16x 3 6x 2 + 4x Multiply each term of the first trinomial by -2x -16x x x 14 Combine like terms

117 Multiplying Polynomials Multiply Polynomials Alg.: D Simplify: (2t + 3)(2t + 3) (t 2)(t 2) 4t 2 + 6t + 6t + 9 [t 2 2t 2t + 4] 4t t + 9 [t 2 4t + 4] 4t t + 9 t 2 + 4t 4 3t t + 5 Use FOIL Combine like terms Distribute negative Combine like terms

118 Linear Equations in One Variable 1. Solving Linear Equations x+b=c and ax=c 2. Solving Linear Equations ax+b=c 3. More Linear Equations ax+b=dx+c

119 Linear Equations Equation expresses the equality of two mathematical expressions Where s the solution? Solution is a number when substituted for the variable results in a true equation

120 Linear Equations If a, b, and c are constants and a 0 then a linear equation in x is an equation that can be written in the form: ax + b = c. (Note: A linear equation in x is also called a first-degree equation in x because the variable x can be written with the exponent 1. That is x = x 1.

121 Solving Linear Equations To solve an equation means to find a solution of the equation. The simplest equation to solve is an equation of the form variable = constant, because the constant is the solution. (ex: x = 5) The objective of solving linear (or first degree) equations is to get the variable by itself (with a coefficient of 1) on one side of the equation and any constants on the other side.

122 Addition Property of Equations The same number or variable term can be added to each side of an equation without changing the solution of the equation. Think of an equation as a balance scale. If the weights added to each side of the equation are not the same, the sides no longer balance. Whatever you do to the one side of the equation, you have to do to the other side to keep it balanced!

123 Solve equations of the form x + a = b x + 5 = x = 4 The goal is to rewrite the equation in the form variable = constant. Add the opposite of the constant term 5 to each side. Simplify

124 Solve equations of the form x + a = b Try: x + 6 = 10 4 = 10 + x x 3 = x = 5 x 4 = 9 3 = x - 6

125 Solve equations of the form x + a = b Section 3.1 Answers: x + 6 = 10 x = 4 4 = 10 + x -6 = x x 3 = x = 5 x = 15 x = 13 x 4 = 9 3 = x 6 x = 13 9 = x

126 Multiplication Property of Equations Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation. The multiplication property is used to remove a coefficient from a variable term in an equation by multiplying each side of the equation by the reciprocal of the coefficient. Still think of an equation as a balance scale. What ever you do to one side of the equation, you must do to the other side to remain equal.

127 Solve equations of the form ax = b 2x = 6 ½ (2x) = ½ (6) x = 3 Multiply each side of the equation by the reciprocal of the coefficient. Simplify to variable = constant. Try: 4x = 8-3x = -9 3x = -27 y/8 = 2 1/3(x) = 5 x/3 = 5

128 Answers: Solve equations of the form ax = b Section 3.1 4x = 8-3x = -9 x = 2 x = 3 3x = -27 y/8 = 2 x = -9 y = 16 1/3(x) = 5 x/3 = 5 x = 15 x = 15

129 Solving Application Problems Section The perimeter of a square is 4 times the length of a side (P = 4s). Find the length of a side if the perimeter of a square is 64 ½ meters. 4s = 64 ½ 4 4 s =

130 Solving Application Problems 2. A suit cost the store \$675 and the owner said that he wants to make a profit of \$180. What price should he mark on the suit? Profit = Revenue Cost 180 = R = R Therefore, the price he would mark on the suit would be \$855.

131 Solving Application Problems 3. How long will a truck driver take to travel 350 miles if he averages 50 mph? Distance = Rate x Time (d = rt) 350 = 50t = t Therefore, it would take 7 hours

132

133 Solving Linear Equations: ax + b = c Section 3.2 The goal is to rewrite the equation in the form variable = constant, because the constant is the solution. This requires applying both the Addition and Multiplication Properties of Equations.

134 Solve equations of the form ax + b = c Section 3.2 3x 7 = x = x = 2/3 Undo (add opposite) any addition or subtraction first. Then undo (multiply reciprocal) any multiplication or division. Simplify

135 Try these 4x + 3 = 11 7 x = 9 8y + 4 = 12 5x 6 = 9-35 = -6x + 1 3x 7 = -5 2 = 5x m 21 = 0 5 = 9 2x (2/5)x 3 = -7 6a a = 11 2x 6x + 1 = 9

136 Try these (answers) 4x + 3 = 11 x = 2 5x 6 = 9 x = 3 2 = 5x + 12 x = -2 (2/5)x 3 = -7 x = x = 9 x = = -6x + 1 x = 6-3m 21 = 0 m = -7 6a a = 11 a = 1 8y + 4 = 12 y = 1 3x 7 = -5 x = 2/3 5 = 9 2x x = 2 2x 6x + 1 = 9 x = -2

137 Applications 1. One hundred eighty-four feet of fencing is needed to enclose a rectangular shaped garden plot. If the plot is 23 feet wide, what is the length? P = 2l + 2w 184 = 2l + 2(23) 184 = 2l = 2l = l Therefore, the length of the fence is 69 feet

138 Solve Linear Equations: ax + b = cx + d Section 3.3 4x 5 = 6x x -6x -2x 5 = x = x = -8 Begin by rewriting the equation so there is only one variable term in the equation. Simplify to variable = constant.

139 Try these 8x + 5 = 4x x 4 = 9x 7 2b + 3 = 5b x + 4 = 6x x 2 = 4x 13 4x 7 = 5x + 1 5x 4 = 2x + 5 2x 3 = -11 2x 4y 8 = y - 8

140 Try these (answers) 8x + 5 = 4x +13 x = 2 7x + 4 = 6x + 7 x = 3 5x 4 = 2x + 5 x = 3 12x 4 = 9x 7 x = -1 15x 2 = 4x 13 x = -1 2x 3 = -11 2x x = -2 2b + 3 = 5b + 12 b = -3 4x 7 = 5x + 1 x = -8 4y 8 = y 8 y = 0

141 Solve equations containing parentheses 4 + 5(2x 3) = 3(4x 1) x 15 = 12x 3 10x 11 = 12x 3-12x -12x -2x - 11= x = x = -4 Begin by using the Distributive Property. Get variable terms on one side. Get constant terms on the other side. Simplify to variable = constant.

142 Try these 6y + 2(2y + 3) = 16 12x 2(4x 6) = 28 9x 4(2x 3) = x = 12 (6x + 7) 5[2 (2x 4)] = 2(5 3x) x + 5(3x 20) = 10(x 4) -4[x + 3(x 5)] = 3(8x + 20) 3[2 4(2x 1)] = 4x 10

143 Try these (answers) 6y + 2(2y + 3) = 16 y = 1 9x 4(2x 3) = 11 x = -1 5[2 (2x 4)] = 2(5 3x) x = 5-4[x + 3(x 5)] = 3(8x + 20) x = 0 12x 2(4x 6) = 28 x = 4 9 5x = 12 (6x + 7) x = -4 x + 5(3x 20) = 10(x 4) x = 10 3[2 4(2x 1)] = 4x 10 x = 1

144 Applications 1. A sail is in the shape of a triangle. Find the height of the sail if its base is 20 ft and its area is 300 ft 2. A = ½bh 300 = ½ (20) h 300 = 10h = h Therefore, the height of the sail is 30 feet.

145 Applications 2. The height of a trapezoid is 10 ft and its area is 250 ft 2. If one of the bases is 2 ft more than 3 times the other base, what are the lengths of the bases? A = ½h(b + c) Let b = the = ½(10)(b + 3b+2) st base Then the 2 nd base would be 3b = 5(4b + 2) 250 = 20b = 20b 12 = b = 1 st base so the 2 nd base = 3(12) + 2 = 38

146 Applications 3. If six times a number is increased by 15.35, the result is 12.5 less than the number. What is the number? Let x = the number 6x = x x -x 5x = x = x = -5.57

147 Linear Equations in Two Variables 1. Intercepts 2. Graph by Plotting Points 3. Slope Intercept Form: y = mx + b 4. Point-Slope Form: y y 1 = m(x x 1 )

148 x-coordinate y-coordinate Ordered Pair (3, 4) Horizontal Distance (left & right) Vertical Distance (up & down)

149 Intercepts The point of intersection with the x-axis is called the x- intercept, the value of x on the coordinate plane where y = 0 (i.e. the point would look like (x, 0)). The point of intersection with the y-axis is called the y- intercept, the value of y on the coordinate plane where x = 0 (i.e. the point would look like (0, y)).

150 Linear Equations FORM Slope = rise/run Vertical Lines Horizontal Lines Slope-intercept EQUATION x = constant y = constant y = mx + b Point-slope y y 1 = m(x x 1 ) Standard Ax + By = C Parallel Lines Have same slope (m 1 =m 2 ) Perpendicular Lines m y x 2 2 y x 1 1 Have negative reciprocal slope (m 1 = -1/m 2 )

151 Linear Equations: Graph by Plotting Points Graph y = 2x + 1 Make a t-chart: By choosing any value for x and substituting that value into the linear equation, we can find the corresponding value of y. Usually we use x = 0 because it is easy to calculate. x y = 2x + 1 y 0 2(0) (1) (-1) Graph the ordered pair solutions (0, 1), (1, 3), and (-1, -1). Draw a line through the ordered-pair solutions.

152 Linear Equations: Slope-Intercept Form Slope-Intercept Form An equation of a line of the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is the measure of its slant. Slope = m = rise/run. The point of intersection with the y-axis is called the y- intercept, the value of y on the coordinate plane where x = 0 (i.e. the point would look like (0, y)).

153 Linear Equations Slope-Intercept Form Example: It is easy to graph a linear equation when it is in slope-intercept form. The following equation will be converted from standard form, -6x + y = -2, into slope-intercept form, y = mx + b. To do so the equation will need to be manipulated so that it is solved for y. Isolate y on the left-hand side by itself. -6x + y = -2 Add 6x to each side y = 6x - 2 Now in Slope-intercept form Identify the slope, which is represented by m in the equation and the y- intercept, which is represented by b in the equation. y = 6x - 2 When using the slope-intercept form to graph, make sure the equation is solved for y BEFORE identifying the slope and y-intercept.

154 Slope = m =rise/run m = ½ So you rise 1 and run 2. Graph a line using the slope-intercept form of a line: 1. Locate the y-intercept and plot the point 2. From this point, use the slope to find the second point and plot. 3. Draw a line that connects the two points.

155 Graph Solutions Example 1: Graph equation: y = ( -2 / 3 )x + 2 You can see from the equation and the graph that the slope of the line is ( -2 / 3 ), and the y-intercept is 2. The graph crosses the y-axis at (0, 2) Rise = -2 (so moving down 2 units) Run = 3 (so moving to the right 3 units)

156 Linear Equations: Point-Slope Form Point-Slope form An equation of a line that has slope m and contains the point whose coordinates are (x 1, y 1 ) can be found by the point-slope formula: y y 1 = m(x x 1 ) Example: Find the equation of a line that passes through the point whose coordinates are (-2, -1) and has slope 3/2. y y y y ( ) 3 ( x 2 3 ( x 2 2) 3 x 2 3 x 2 ( 2)) Substitute values Distribute Subtract 1 from both sides

157 Example: Find the equation of the line that passes through (1, 2) and (-3, 4). Step 1: Find the slope. Let x 1 = 1 and y 1 = 2. Let x 2 = -3 and y 2 = 4. m y x 2 2 y x Step 2: Substitute the slope (m) and a point, (1, 2) in the slope-intercept form of a line. Solve for the y-intercept (b). Substitute: x = 1, y = 2, and m = - ½ into y = mx + b and solve for b. If 2 = - ½ (1) + b then b = 5/2 Step 3: Substitute the slope (m) and the y- intercept (b) into the slope-intercept form of a line, y = mx + b. y = - ½ x + 5/2 OR use Point-Slope Formula y y y x ( x x 5 2 1) 1 2

158 Horizontal Lines y = 1 (or any number) Lines that are horizontal have a slope of zero. They have "run", but no "rise". The rise/run formula for slope always yields zero since rise = 0. y = mx + b y = 0x + 1 y = 1 This equation also describes what is happening to the y- coordinates on the line. In this case, they are always 1. Vertical Lines x = -1 (or any number) Lines that are vertical have no slope (it does not exist). They have "rise", but no "run". The rise/run formula for slope always has a zero denominator and is undefined. These lines are described by what is happening to their x- coordinates. In this example, the x-coordinates are always equal to -1. x = -1 y = 1

159 Slope of a Line The slope of a line is the measure of its slant. The symbol for slope is m. Slope = m = rise/run = Positive slope: slants upward Negative slope: slants downward Horizontal = 0 slope Vertical = undefined By knowing this information, you should have an idea of what your graph will look like before you plot any points. See

160 Try These: Graph the following lines: y = ¾ x + 2 y = 3x 4 y = -⅓ x + 2 y = -2x + 3 y = -⅓ x + 4 2x + y = 4 x 3y = 6 y = -4 x = -8

161

162

163 Linear Equations: Parallel and Perpendicular Lines Parallel lines - Lines in the same plane that do not cross, the distance between the lines is constant. Two different, nonvertical lines with slopes m 1 and m 2 are parallel if and only if they have the same slope (m 1 = m 2 ). Perpendicular lines - Lines that intersect at one point forming 90 angles. Two different, nonvertical lines with slopes m 1 and m 2 are perpendicular if and only if have their slopes are negative reciprocals of each other (m 1 = -1/m 2 ). The product of their slopes is (-1).

164 Parallel Lines Parallel lines - Lines in the same plane that do not cross, the distance between the lines is constant. Two different, nonvertical lines with slopes m 1 and m 2 are parallel if and only if they have the same slope (m 1 = m 2 ). Determine if y = 2x + 5 and -2x + y = -2 are parallel. Equation Slope- Intercept Slope y = 2x + 5 y = 2x x + y = -2 y = 2x 2 2 Since the slopes are the same (equal), the two lines are parallel.

165 Parallel Lines Example: Write the equation of the line that passes through the point Q(3,7) and is parallel to the line y = 6x + 6. Since the goal is to write an equation that is parallel to the line y = 6x + 6, the equations will have the same slope (m). The given equation is written in the slopeintercept form, therefore m = 6. To write an equation with m = 6 and passing through the point Q, the point-slope form is used. y - y 1 = m(x - x 1 ) For point Q(3,7), x 1 = 3 and y 1 = 7. y + 7 = 6(x + 3) y - 7 = 6(x - 3) Distribute the 6. (multiply) y - 7 = 6x - 18 Add 7 to each side y = 6x 11 Therefore, line y = 6x 11 passes through point Q(3,7) and is parallel to line y = 6x + 6.

166 Perpendicular Lines Perpendicular lines - Lines that intersect at one point forming 90 angles. Two different, nonvertical lines with slopes m 1 and m 2 are perpendicular if and only if have their slopes are negative reciprocals of each other (m 1 = -1/m 2 ). The product of their slopes is (-1). Determine if y = 2x + 5 and ½ x + y = -2 are perpendicular. Equation Slope- Intercept Slope y = 2x + 5 y = 2x ½ x + y = -2 y = - ½ x 2 - ½ Since the slopes are negative reciprocals of each other, the two lines are perpendicular.

167 Perpendicular Lines Example: Write the equation of the line that passes through the point V(3,11) and is perpendicular to the line y = -3/4x Since the goal is to write an equation that is perpendicular to the line y = -3/4x + 24, the equations will have slopes (m) that are negative reciprocals. The given equation is written in the slope-intercept form, therefore m = -3/4. The negative reciprocal is 4/3. To write an equation with m = 4/3 and passing through the point V, the pointslope form is used. y - y 1 = m(x - x 1 ) For point V(3,11), x 1 = 3 and y 1 = 11. y y y x 4 ( x 3 4 x ) Distribute the 4/3. (multiply) Add 11 to each side. Therefore, line y = (4/3)x + 15 passes through point V(3,11) and is perpendicular to line, y = (-3/4)x + 24.

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

### Sample Test Questions

mathematics Numerical Skills/Pre-Algebra Algebra Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about

More information

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

More information

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

### MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006

MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order

More information

### Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond

More information

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

### Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

More information

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

### Algebra 2 PreAP. Name Period

Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

### Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

More information

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

### Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable

More information

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

More information

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

More information

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

More information

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

More information

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

More information

### IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

More information

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

### MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

### Algebra 1. Curriculum Map

Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring

More information

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

More information

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

More information

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

### FACTORING POLYNOMIALS

296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

### 1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

More information

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

More information

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

More information

### CAMI Education linked to CAPS: Mathematics

- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

More information

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

### CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses

More information

### Answers to Basic Algebra Review

Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

More information

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

More information

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

### Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

More information

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

More information

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

More information

### Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

More information

### Prentice Hall Mathematics, Algebra 1 2009

Prentice Hall Mathematics, Algebra 1 2009 Grades 9-12 C O R R E L A T E D T O Grades 9-12 Prentice Hall Mathematics, Algebra 1 Program Organization Prentice Hall Mathematics supports student comprehension

More information

### Algebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

More information

### ( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

More information

### COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

### Blue Pelican Alg II First Semester

Blue Pelican Alg II First Semester Teacher Version 1.01 Copyright 2009 by Charles E. Cook; Refugio, Tx (All rights reserved) Alg II Syllabus (First Semester) Unit 1: Solving linear equations and inequalities

More information

### Solving Quadratic Equations

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

### 7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

### FACTORING OUT COMMON FACTORS

278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information

### BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

More information

### MATD 0390 - Intermediate Algebra Review for Pretest

MATD 090 - Intermediate Algebra Review for Pretest. Evaluate: a) - b) - c) (-) d) 0. Evaluate: [ - ( - )]. Evaluate: - -(-7) + (-8). Evaluate: - - + [6 - ( - 9)]. Simplify: [x - (x - )] 6. Solve: -(x +

More information

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

More information

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

More information

### Factoring Quadratic Expressions

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

### Factor Polynomials Completely

9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

More information

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

More information

### This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

### Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

More information

### Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

### A Systematic Approach to Factoring

A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

More information

### By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.

SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor

More information

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

More information

### How To Solve Factoring Problems

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

More information

### Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

### ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

More information