Algebra Cheat Sheets

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1 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 as they are presented in the Algebra Class Software. Concept Cheat Sheet Adding Integers 1 Subtracting Integers 2 Multiplying Integers 3 Dividing Integers 4 Absolute Value 5 Combining Like Terms 6 Distributive Property I 7 Adding Expressions 8 Subtracting Expressions 9 Writing Expressions 10 Evaluating Expressions 11 Solving Equations 12 Solving Inequalities 13 Writing Equations 14 Writing Inequalities 15 Solving Literal Equations 16 Points on the Coordinate Plane 17 Graphing Using Slope and Intercept 18 Graphing Using Function Tables 19 Find the Slope of a Line from Two Points 20 Equation of a Line 21 Graphing Inequalities 22 Copyright 2004 Senari Programs AlgCS_TOC.doc

2 Sheets Concept Cheat Sheet Forms of Linear Equations 23 Solving Equations II 24 Multiplying Monomials 25 Dividing Monomials 26 Raising Monomials to a Power 27 Negative Powers of Monomials 28 Dividing by a Monomial 29 Greatest Common Factor (GCF) 30 Combining Like Terms II 31 Adding Polynomials 32 Subtracting Polynomials 33 Missing Factors 34 Degree of a Polynomial 35 Multiplying Polynomials by 1 36 Multiplying a Polynomial by a Variable 37 Multiplying a Polynomial by an Integer 38 Multiplying a Polynomial by a Monomial 39 Multiplying Two Binomials 40 Copyright 2004 Senari Programs AlgCS_TOC.doc

3 Sheet 1 Adding Integers Adding means combining 1. If the signs are the same, then add and use the same sign. 2. If the signs are different, then subtract and use the sign of the larger number = = = = 4 Adding Integers Examples

4 Sheet 2 Subtracting Integers Subtracting is the opposite of adding. Change the sign of the second term and add. 8 (+4) = 8 4 = 4 8 (+4) = 8 4 = 12 8 ( 4) = = 12 8 ( 4) = = 4 Subtracting Integers Examples

5 Sheet 3 Multiplying Integers Multiply integers as you would whole numbers, then apply the sign rules to the answer. 1. If the signs are the same, the product is positive. 2. If the signs are different, the product is negative. Multiplying Integers Examples (8) (4) = 32 ( 8) ( 4) = 32 ( 8) (4) = 32 (8) ( 4) = 32

6 Sheet 4 Dividing Integers Divide integers as you would whole numbers, then apply the sign rules to the answer. 1. If the signs are the same, the quotient is positive. 2. If the signs are different, the quotient is negative = = = = 4 Dividing Integers Examples

7 Sheet 5 Absolute Value The absolute value of a number is the distance a number is from 0 on the number line. The absolute values of 7 and -7 are 7 since both numbers have a distance of 7 units from 0. Absolute Value Examples

8 Sheet 6 Combining Like Terms To combine terms, the variables must be identical. 1. Put the terms in alphabetical order. 2. Combine each set of like terms. 3. Put the answers together. Combining Like Terms Examples 3a + 4b + 2c + 5a 6c 2b 1. Put the terms in alphabetical order: 3a + 5a + 4b 2b + 2c 6c 2. Combine each set of like terms: 3a + 5a = 8a 4b 2b = 2b 2c 6c = 4c 3. Put the answer together: 8a + 2b 4c

9 Sheet 6 Combining Like Terms To combine terms, the variables must be identical. 1. Put the terms in alphabetical order. 2. Combine each set of like terms. 3. Put the answers together. Combining Like Terms Examples

10 Sheet 7 Distributive Property I Multiply each term inside the parenthesis by the term on the outside of the parenthesis. c(a + b) = ca + cb c(a - b) = ca - cb Distributive Property Examples 3b (a + 4) = 3b (a) + 3b (4) = 3ab + 12b

11 Sheet 7 Distributive Property I Multiply each term inside the parenthesis by the term on the outside of the parenthesis. c(a + b) = ca + cb c(a - b) = ca - cb Distributive Property Examples

12 Sheet 8 Adding Expressions 1. Set the problem up vertically. 2. Combine the terms. Adding Expressions Examples Write the problem vertically: 8c + 2d - 4g -7c + 4d - 8g 1. Combine the c s: 8c - 7c = c 2. Combine the d s: 2d + 4d = 6d 3. Combine the g s: -4g - 8g = -12g 4. The answer is: c + 6d - 12g

13 Sheet 8 Adding Expressions 1. Set the problem up vertically. 2. Combine the terms. Adding Expressions Examples

14 Sheet 9 Subtracting Expressions 3. Set the problem up vertically. 4. Change the signs of each term on the bottom line. 5. Combine the terms. Subtracting Expressions Examples Change the bottom signs: 8c + 2d - 4g - 7c - 4d + 8g Combine the terms: 8c + 2d - 4g -7c + 4d - 8g 1. Combine the c s: 8c - 7c = c 2. Combine the d s: 2d + 4d = 6d 3. Combine the g s: -4g - 8g = -12g 4. The answer is: c + 6d - 12g

15 Sheet 9 Subtracting Expressions 1. Set the problem up vertically. 2. Change the signs of each term on the bottom line. 3. Combine the terms. Subtracting Expressions Examples

16 Sheet 10 Writing Expressions Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the expression in the order that the words appear. Writing Expressions Examples 1. The product of 4 and x The product of y and 5 2. x more than three thirteen less than y 3. the sum of ten and x the difference between y and 4 4x 5y 3 + x y x y 4

17 Sheet 10 Writing Expressions Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the expression in the order that the words appear. Writing Expressions Examples

18 Sheet 11 Evaluating Expressions Step 1. Replace the variable with parentheses. Step 2. Place the value of the variable inside the parentheses. Step 3. Calculate the answer. Evaluating Expressions Examples Evaluate 10x + 7, when x = 5. Step ( ) + 7 Step (5) + 7 Step = 57

19 Sheet 11 Evaluating Expressions Step 1. Replace the variable with parentheses. Step 2. Place the value of the variable inside the parentheses. Step 3. Calculate the answer. Evaluating Expressions Examples

20 Sheet 12 Solving Equations Step 1. Get all the variables on the left and all the numbers on the right of the equal sign by adding opposites. Step 2. Divide by the coefficient of the variable to determine its value. Solving Equations Examples 2d + 3 = = 3 2d = d = 5

21 Sheet 12 Solving Equations Step 1. Get all the variables on the left and all the numbers on the right of the equal sign by adding opposites. Step 2. Divide by the coefficient of the variable to determine its value. Solving Equations Examples

22 Sheet 13 Solving Inequalities Step 3. Get all the variables on the left and all the numbers on the right of the sign by adding opposites. Step 4. Divide by the positive value of the variable s coefficient. Step 5. If the variable is negative, divide by 1 and reverse the sign. Solving Inequalities Examples 2d + 3 < = 3 2d < d < 5 3. d > 5

23 Sheet 13 Solving Inequalities Step 1. Get all the variables on the left and all the numbers on the right of the sign by adding opposites. Step 2. Divide by the positive value of the variable s coefficient. Step 3. If the variable is negative, divide by 1 and reverse the sign. Solving Inequalities Examples

24 Sheet 14 Writing Equations Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the equation in the order that the words appear. 4. The equal sign is used in place of the word is. Writing Equations Examples 1. The product of 4 and x is 12. The product of y and 5 is x more than three is 12. Thirteen less than y is The sum of ten and x is 12. The difference between y and 4 is 2. 4x = 12 5y = x = 12 y 13 = x = 12 y 4 = 2

25 Sheet 14 Writing Equations Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the equation in the order that the words appear. 4. The equal sign is used in place of the word is. Writing Equations Examples

26 Sheet 15 Writing Inequalities Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the equation in the order that the words appear. 4. The < is used in place of is less than. 5. The > is used in place of is greater than. Writing Inequalities Examples 1. The product of 4 and x is greater than x more than three is less than The difference between y and 4 is greater than 2. 4x > x < 12 y 4 > 2

27 Sheet 15 Writing Inequalities Look for clue words: 1. For the clue words, the product of place the constant before the variable. Do not use a sign. 2. The clue words more than and less than indicate inverted order. 3. If there are no clue words, write the equation in the order that the words appear. 4. The < is used in place of is less than. 5. The > is used in place of is greater than. Writing Inequalities Examples

28 Sheet 16 Solving Literal Equations Step 1. Get the desired variable on the left and all the others on the right of the equal sign by adding opposites. Step 2. Divide both sides by the positive value of any other variable on the left. Solving Literal Equations Example Solve for l (length) A = lw Divide by w 1. lw = A 2. A l = w

29 Sheet 16 Solving Literal Equations Step 1. Get the desired variable on the left and all the others on the right of the equal sign by adding opposites. Step 2. Divide both sides by the positive value of any other variable on the left. Solving Literal Equations Example

30 Sheet 17 Locating Points: Step 1. Find the location on the x-axis. It is 3. Step 2. Find the location on the y-axis. It is 4. Step 3. Write the location in this form (x, y). The point is ( 3, 4). Points on the Coordinate Plane Plotting points: Plot the point (5, 3) on the coordinate plane. Step 1. Begin at point (0, 0). Move 5 to the right (on the x-axis) since 5 is positive. Step 2. Move 3 down since 3 is negative. Step 3. Plot the point. Notes Points on the Coordinate Plane Use graph paper. Begin by marking the x-axis and y-axis as shown in the diagram above.

31 Sheet 18 Graphing Using Slope and Intercept The y-intercept is the constant in the equation. It is the value of y when x = 0. A line with a positive slope goes up and to the right. A line with a negative slope goes down and to the right. To graph the equation: y = 5x 3 Step 1. The y-intercept is 3. Plot the point (0, 3). Step 2. The slope is 5. Move 1 to the right and 5 up from the first point. Plot the second point at (1, 2). Step 3. Draw the line connecting the two points. Graphing - Using the Slope and y-intercept y = 5x 3

32 Sheet 18 Graphing Using Slope and Intercept The y-intercept is the constant in the equation. It is the value of y when x = 0. A line with a positive slope goes up and to the right. A line with a negative slope goes down and to the right. To graph the equation: y = 5x 3 Step 1. The y-intercept is 3. Plot the point (0, 3). Step 2. The slope is 5. Move 1 to the right and 5 up from the first point. Plot the second point at (1, 2). Step 3. Draw the line connecting the two points. Graphing - Using the Slope and y-intercept

33 Sheet 19 Graphing Using Function Tables Step 1. Substitute 0 for x, and calculate the value of y. Enter both on the first line of the function table. Step 2. Substitute 1 for x, then calculate the value of y. Enter both numbers on the second line of the function table. Step 3. Plot the two points, and draw a line between them. Graphing - Using Function Tables y = 5x 3 Step 1. Substitute 0 for x; y = -3. Enter x and y on the first line. Step 2. Substitute 1 for x; y = 2. Enter these numbers on the second. Step 3. Plot the two points, (0, -3) and (1, 2) then draw a line between them. x y

34 Sheet 19 Graphing Using Function Tables Step 1. Substitute 0 for x, and calculate the value of y. Enter both on the first line of the function table. Step 2. Substitute 1 for x, then calculate the value of y. Enter both numbers on the second line of the function table. Step 3. Plot the two points, and draw a line between them. Graphing - Using Function Tables

35 Sheet 20 Find the Slope of a Line from Two Points Step 1. Make a function table, entering the x and y values of the two points. Step 2. Subtract: y1 y2 = Rise Step 3. Subtract: x1 x2 = Run Step 4. Slope = Rise Run = y1 y2 x1 x2 Slope of a Line from Two Points Find the slope of the line which includes points (-2, -2) and (1, 6). Step 1. Enter the x and y values for the two points. Step 2. The rise is (-2) (5) or 7. Step 3. The run is (-2) (1) or 1 Step 4. The slope is 4. x y Run Rise Rise 8 = Run 2

36 Sheet 20 Find the Slope of a Line from Two Points Step 1. Make a function table, entering the x and y values of the two points. Step 2. Subtract: y1 y2 = Rise Step 3. Subtract: x1 x2 = Run Step 4. Slope = Rise Run = y1 y2 x1 x2 Slope of a Line from Two Points

37 Sheet 21 Equation of a Line To find the equation of a line shown on a graph. Step 1. Determine the y-intercept. Find the point where x = 0. Substitute this value in the equation in place of b. Step 2. Find the value of y when x = 1. Step 3. Subtract the value found in Step 1 from the value found in Step 2. This is the slope. Substitute this value in the equation in place of m. Equation of a Line Example Step 1. The y-intercept is 6. y = mx + 6. Step 2. When x = 1, y = 2. Step 3. The slope (m) is 2 6 or 4. The equation of the line is: y = -4x + y

38 Sheet 21 Equation of a Line To find the equation of a line shown on a graph. Step 1. Determine the y-intercept. Find the point where x = 0. Substitute this value in the equation in place of b. Step 2. Find the value of y when x = 1. Step 3. Subtract the value found in Step 1 from the value found in Step 2. This is the slope. Substitute this value in the equation in place of m. Equation of a Line Example

39 Sheet 22 Graphing Inequalities Step 1. Plot the y-intercept. Step 2. Use the slope as the rise, and 1 as the run. Count the rise and run to find another point. Step 3. Determine what kind of a line will connect the points. (Use a dotted line for < or >; use a solid line for or.) Step 4. Shade above the line for greater than (>), and below the line for less than (<). Example Graphing Inequalities y x + 2 Step 1. Substitute 0 for x; y = 2. Enter x and y on the first line. Step 2. Substitute 1 for x; y = 3. Enter these numbers on the second. Step 3. Plot the two points, (0, 2) and (1, 3) then draw a solid line between them. Step 4. Shade below the line because the sign is.

40 Sheet 22 Graphing Inequalities Step 1. Plot the y-intercept. Step 2. Use the slope as the rise, and 1 as the run. Count the rise and run to find another point. Step 3. Determine what kind of a line will connect the points. (Use a dotted line for < or >; use a solid line for or.) Step 4. Shade above the line for greater than (>), and below the line for less than (<). Example Graphing Inequalities

41 Sheet 23 Slope intercept form Standard form Forms of Linear Equations y = mx + b Ax + by = C Forms of Linear Equations Examples Change the slope-intercept equation to standard form. 2 y = 3 x 3 slope-intercept form 5y = 2x 15 Multiply each side by 5. 2x + 5y = 15 Subtract 2x from each side. Change the standard form equation to slopeintercept form. 3x + 2y = 6 Standard form equation 2y = 3x + 6 Subtract 3x from each side. 3 y = 2 x + 3 Slope-intercept form Copyright 1999 Senari Programs alg_cs_1.doc

42 Sheet 23 Slope intercept form Standard form Forms of Linear Equations y = mx + b Ax + by = C Forms of Linear Equations Examples Copyright 1999 Senari Programs alg_cs_1.doc

43 Sheet 24 Solving Equations II Step 1. Get all the variables on the left and all the numbers on the right of the equal sign by adding opposites. Step 2. Combine like terms. Step 3. Divide by the coefficient of the variable to determine its value. Solving Equations II Example 4x + 4 = 2x x 2x = x = x = 5 Copyright 1999 Senari Programs alg_cs_1.doc

44 Sheet 24 Solving Equations II Step 1. Get all the variables on the left and all the numbers on the right of the equal sign by adding opposites. Step 2. Combine like terms Step 3. Divide by the coefficient of the variable to determine its value. Solving Equations II Examples Copyright 1999 Senari Programs alg_cs_1.doc

45 Sheet 25 Multiplying Monomials To multiply monomials, add the exponents of the same variables. Example Multiplying Monomials Step 1. Multiply the a s Step 2. Multiply the b s Step 3. Multiply the c s Step 4. Put them together: Copyright 1999 Senari Programs alg_cs_1.doc

46 Sheet 25 Multiplying Monomials To multiply monomials, add the exponents of the same variables. Examples Multiplying Monomials Copyright 1999 Senari Programs alg_cs_1.doc

47 Sheet 26 Dividing Monomials To divide monomials, subtract the exponents of the same variables. Example Dividing Monomials Step 1. Divide the a s Step 2. Divide the b s Step 3. Divide the c s Step 4. Put them together: Copyright 1999 Senari Programs alg_cs_2.doc

48 Sheet 26 Dividing Monomials To divide monomials, subtract the exponents of the same variables. Examples Dividing Monomials Copyright 1999 Senari Programs alg_cs_2.doc

49 Sheet 27 Raising Monomials to a Power To raise monomials to a power, multiply the exponent of each variable by the power. Example Raising Monomials to a Power Step 1. Square the a s Step 2. Square the b s Step 3. Square the c s Step 4. Put them together: Copyright 1999 Senari Programs alg_cs_2.doc

50 Sheet 27 Raising Monomials to a Power To raise monomials to a power, multiply the exponent of each variable by the power. Examples Raising Monomials to a Power Copyright 1999 Senari Programs alg_cs_2.doc

51 Sheet 28 Negative Powers of Monomials To multiply, divide or raise monomials to negative powers, use the rules for integers. Example Raising Monomials to a Power Step 1. Multiply the m s Step 2. Multiply the n s Step 3. Multiply the p s Step 4. Put them together: Copyright 1999 Senari Programs alg_cs_2.doc

52 Sheet 28 Negative Powers of Monomials To multiply, divide or raise monomials to negative powers, use the rules for integers. Examples Raising Monomials to a Power Copyright 1999 Senari Programs alg_cs_2.doc

53 Sheet 29 Dividing by a Monomial To divide by a monomial, separate the given expression into the sum of two fractions and divide. Dividing by a Monomial Example 12x 2 8x 4x 12x 2 8x + = 3x 2 4x 4x Copyright 1999 Senari Programs alg_cs_2.doc

54 Sheet 29 Dividing by a Monomial To divide by a monomial, separate the given expression into the sum of two fractions and divide. Dividing by a Monomial Example Copyright 1999 Senari Programs alg_cs_2.doc

55 Sheet 30 Greatest Common Factor The greatest common factor (GCF) is the greatest number that is a factor of two or more given numbers. In algebra, the GCF consists of the GCF of the coefficients multiplied by the GCF of the variables. Greatest Common Factor Example 6x 2 y + 3xy 2 The GCF of the coefficients, 6 and 3, is 3. The GCF of the variables, x 2 y and xy 2 is xy The product is 3xy; this is the GCF. Copyright 1999 Senari Programs alg_cs_2.doc

56 Sheet 30 Greatest Common Factor The greatest common factor (GCF) is the greatest number that is a factor of two or more given numbers. In algebra, the GCF consists of the GCF of the coefficients multiplied by the GCF of the variables. Greatest Common Factor Examples Copyright 1999 Senari Programs alg_cs_2.doc

57 Sheet 31 Combining Like Terms II Step 1. Arrange the terms in descending order of exponents. Step 2. Combine the terms with like exponents and variables. Example Combining Like Terms II -3y 8 + 6y 9 4y 9 + 8y + 7y 8 2y Step 1. Step 2. 6y 9 4y 9 3y 8 + 7y 8 + 8y 2y 2y 9 + 4y 8 + 6y Copyright 1999 Senari Programs alg_cs_2.doc

58 Sheet 31 Combining Like Terms II Step 1. Arrange the terms in descending order of exponents. Step 2. Combine the terms with like exponents and variables. Examples Combining Like Terms II Copyright 1999 Senari Programs alg_cs_2.doc

59 Sheet 32 Adding Polynomials Step 1. Set up the problem vertically in descending order of exponents. Step 2. Add the like terms. Example Adding Polynomials -3y 8 + 6y 9 4y 9 + 8y + 7y 8 2y Step 1. Step 2. 6y 9 4y 9 3y 8 + 7y 8 + 8y 2y 2y 9 + 4y 8 + 6y Copyright 1999 Senari Programs alg_cs_2.doc

60 Sheet 32 Adding Polynomials Step 1. Set up the problem vertically in descending order of exponents. Step 2. Add the like terms. Examples Adding Polynomials Copyright 1999 Senari Programs alg_cs_2.doc

61 Sheet 33 Subtracting Polynomials Step 1. Set up the problem vertically in descending order of exponents. Step 2. Change the signs of all the bottom terms. (the ones to be subtracted) Step 3. Combine the like terms. A short way of saying this is: Change the bottom signs and add. Example Subtracting Polynomials (-20x x x) (30x x x) Step 1. 30x 3 20 x x (10x x 2 20x + 2) Step 2. 30x 3 20 x x x 3 30x x 2 Step 3. 20x 3 50 x x + 1 Copyright 1999 Senari Programs alg_cs_2.doc

62 Sheet 33 Subtracting Polynomials Step 1. Set up the problem vertically in descending order of exponents. Step 2. Change the signs of all the bottom terms. (the ones to be subtracted) Step 3. Combine the like terms. A short way of saying this is: Change the bottom signs and add. Examples Subtracting Polynomials Copyright 1999 Senari Programs alg_cs_2.doc

63 Sheet 34 Missing Factors Missing Factor problems can be set up as multiplication problems with one factor blank, or as a division problem. In either case, the answer is always the same. Missing Factors Example Here are the two ways a problem can be written: The missing factor is x. Copyright 1999 Senari Programs alg_cs_2.doc

64 Sheet 34 Missing Factors Missing Factor problems can be set up as multiplication problems with one factor blank, or as a division problem. In either case, the answer is always the same. Missing Factors Examples Copyright 1999 Senari Programs alg_cs_2.doc

65 Sheet 35 Degree of a Polynomial The degree of a polynomial is highest degree (exponent) of any of its terms after it has been simplified. Example Degree of a Polynomial 40x x 2 + 5x is the largest exponent this is a third degree polynomial Copyright 1999 Senari Programs alg_cs_2.doc

66 Sheet 35 Degree of a Polynomial The degree of a polynomial is highest degree (exponent) of any of its terms after it has been simplified. Examples Degree of a Polynomial Copyright 1999 Senari Programs alg_cs_2.doc

67 Sheet 36 Multiplying Polynomials by -1 To multiply a polynomial by 1, change the sign of each term of the polynomial. Multiplying Polynomials by 1 Examples Copyright 1999 Senari Programs alg_cs_2.doc

68 Sheet 37 Multiply a Polynomial by a Variable Step 1. Multiply each term of the polynomial by the variable. Step 2. Combine the results. Example Multiply Polynomials by Monomials x(2x 2 + 3x 4) Step 1. x(2x 2 ) = 2x 3 x(3x) = 3x 2 x ( 4) = 4x Step 2. Therefore: 2x 3 + 3x 2 4x x(2x 2 + 3x 4) = 2x 3 + 3x 2 4x Copyright 1999 Senari Programs alg_cs_2.doc

69 Sheet 37 Multiply a Polynomial by a Variable Step 1. Multiply each term of the polynomial by the variable. Step 2. Combine the results. Examples Multiply Polynomials by Monomials Copyright 1999 Senari Programs alg_cs_2.doc

70 Sheet 38 Multiply a Polynomial by an Integer Step 1. Multiply each term of the polynomial by the integer. Step 2. Combine the results. Example Multiply a Polynomial by an Integer -2 ( 2x 2 + 3x 4) Step (2x 2 ) = 4x 2-2 (3x) = 6x -2 ( 4) = 8 Step 2. 4x 2 6x + 8 Therefore: -2 ( 2x 2 + 3x 4) = 4x 2 6x + 8 Copyright 1999 Senari Programs alg_cs_2.doc

71 Sheet 38 Multiply a Polynomial by an Integer Step 1. Multiply each term of the polynomial by the integer. Step 2. Combine the results. Examples Multiply a Polynomial by an Integer Copyright 1999 Senari Programs alg_cs_2.doc

72 Sheet 39 Multiply a Polynomial by a Monomial Step 1. Multiply each term of the polynomial by the monomial. Step 2. Combine the results. Example Multiply Polynomials by Monomials -2x ( 2x 2 + 3x 4) Step 1. -2x (2x 2 ) = 4x 3-2x (3x) = 6x 2-2x ( 4) = 8x Step 2. Therefore: 4x 3 6x 2 + 8x -2x(2x 2 + 3x 4) = 4x 3 6x 2 + 8x Copyright 1999 Senari Programs alg_cs_2.doc

73 Sheet 39 Multiply a Polynomial by a Monomial Step 1. Multiply each term of the polynomial by the monomial. Step 2. Combine the results. Examples Multiply Polynomials by Monomials Copyright 1999 Senari Programs alg_cs_2.doc

74 Sheet 40 Multiplying Two Binomials The product of 2 binomials has 4 terms. To find these 4 terms, multiply each term in the 1 st binomial with each term in the 2 nd binomial. This process is called the FOIL method. The formula is: (a + b) (c + d) = ac + ad + bc + bd F = (ac) is the product of the FIRST terms O = (ad) is the product of the OUTSIDE terms I = (bc) is the product of the INSIDE terms L = (bd) is the product of the LAST terms Example Multiplying Two Binomials (m + 3) (m + 4) Step 1. Multiply first (m) (m) = m terms: 2 Step 2. Multiply outside terms: Step 3. Multiply inside terms: Step 4. Multiply last terms Step 5. Combine like terms: (m) (4) = 4m (3) (m) = 3m (4) (3) = 12 m 2 + 4m + 3m + 12 m 2 + 7m + 12 Copyright 1999 Senari Programs alg_cs_2.doc

75 Sheet 40 Multiplying Two Binomials The product of 2 binomials has 4 terms. To find these 4 terms, multiply each term in the 1 st binomial with each term in the 2 nd binomial. This process is called the FOIL method. The formula is: (a + b) (c + d) = ac + ad + bc + bd F = (ac) is the product of the FIRST terms O = (ad) is the product of the OUTSIDE terms I = (bc) is the product of the INSIDE terms L = (bd) is the product of the LAST terms Examples Multiplying Two Binomials Copyright 1999 Senari Programs alg_cs_2.doc

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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

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

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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.

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

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b. PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

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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

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

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,

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Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

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,

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) = Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

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

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What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

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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

( ) 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

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Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

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

A synonym is a word that has the same or almost the same definition of Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

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7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

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15.1 Factoring Polynomials LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE

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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

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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/

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

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

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Factoring - Greatest Common Factor 6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.

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Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown. Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

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is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

Radicals - Rationalize Denominators 8. Radicals - Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens

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In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting

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

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring 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 (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