P.E.R.T. Math Study Guide
|
|
- Melanie Joseph
- 7 years ago
- Views:
Transcription
1 A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide
2 PERT - A Math Study Guide 1. Linear Equations and Inequalities An equality involving an unknown variable of degree one is called a linear equation. Example: 3x + 1 = (x 3) x = 13 4x (x + 5) = 1x 11 Any inequality in which a variable of degree one is included is called a linear inequality. Example: 3x 9 > x < 3x 11, 4x + 3 9x 6(x + 8) x To solve a linear equation: 1. Distribute to remove all ( ). We can add or subtract any number or any algebraic term from both sides of an equation or an inequality. 3. We can multiply or divide both sides of an equation by any number. In the case of inequality, if the number is negative we must change the direction of the inequality sign.. Quadratic Equations Generally, quadratic equations are in the form of ax + bx + c = 0. To solve such equations we use the following formula. We simply replace a, b, and c in the formula, and simplify the numerical expression. b b 4ac x a 3. Literal Equations When a relationship between an unknown quantity and some numerical facts is stated using words instead of algebraic symbols, it is called a verbal equation. To translate a verbal equation to an algebraic equation, we use the following translation rules: Word or Phrase And, plus, more than, total of, sum of, added, increased by Fewer, less, minus, less than, decreased by, subtracted from, difference between Increased by certain times, certain times, multiplied by, product of Ratio of two quantities, quotient of two quantities, a portion out of a whole Algebraic Operation Addition Subtraction Multiplication Division All Rights Reserved
3 PERT - A Math Study Guide 4. Evaluating Algebraic Expressions Evaluating algebraic expressions is a process by which we replace each variable or parameter with its numerical value. In case, the expression contains exponents and terms with multiple factors, we must use parentheses to guard their position in the expression. For example, if we want to evaluate 3ab c 3 for a = 1, b =, and c = 3, we use parentheses as shown below to guard the role of each factor and its exponent: 3( 1)( ) )( 3) 3 5. Factoring Changing the form of an algebraic expression to the product of some other algebraic expressions is called factorization. We use two methods for factoring the algebraic expressions: a. Grouping Method: In this method, we arrange terms of an expression in different groups such that we can create a common factor among the groups for factoring out. For example, we can rearrange the expression m + n 3 + mn + mn by grouping in pairs: m + n 3 + mn + mn = (m + mn ) + (mn + n 3 ) Now, we can factor out the common terms among the elements of the expression twice in order to change it into a factorized form: m + n 3 + mn + mn = (m + mn ) + (mn + n 3 ) Arrange terms in two groups. = m(m + n ) + n(m + n ) Factor out m among the first group Factor out n among the second group. = (m + n )(m + n) Factor out (m + n ) between both groups b. Using Identities: When an algebraic expression is in the form of one of the following algebraic identities, then we can factor the expression directly using the format of the identities. Algebraic Identities: (a + b) = a + ab + b (a b) = a ab + b (a + b) 3 = a 3 + 3a b + 3ab + b 3 (a b) 3 = a 3 3a b + 3ab b 3 (a + b)(a ab + b ) = a 3 + b 3 (a b)(a + ab + b ) = a 3 + b 3 (x + a)(x + b) = x + (a + b)x + ab All Rights Reserved
4 PERT - A Math Study Guide 6. Simplifying Polynomials Simplifying any polynomial involves one simple operation: combining like terms in the polynomial. To combine like terms, we simply combine their numerical coefficients. Example: 3a b 4 c 3, 7a b 4 c 3, and 5a b 4 c 3 To combine these terms, we first combine their coefficients to get 9 Then replace all these terms with only one term with coefficient 9 The simplification of these three terms is: 9a b 4 c 3 7. Adding Polynomials Adding two or more polynomials is simple and does not require any extra operations. Simply write the polynomials horizontally or vertically and combine the similar terms among all the polynomials. 8. Subtracting Polynomials To subtract a polynomial (subtrahend) from another polynomial (minuend), write the minuend and then place the subtrahend inside parentheses with a negative sign before it. Then apply the negative sign to the polynomial inside the parentheses, and simplify. Example: To subtract a b ab + a b from a b 3ab + a b Follow these steps: a b 3ab + a b (a b ab + a b) = a b 3ab + a b a b + ab a + b = a b ab 9. Multiplying Polynomials To multiply a polynomial by another polynomial we use the Distributive Property. Distributive Property: a(m + n + p) = am + an + aq All Rights Reserved
5 PERT - A Math Study Guide Example: To multiply (ab + bc + ac) by (m + n + q ) We multiply each term of the first polynomial by the second polynomial first: (ab + bc + ac)(m + n + q ) = (ab)(m + n + q ) + (bc)(m + n + q ) + (ac)(m + n + q ) Then we distribute each term over the second polynomial using the Distributive Property: (ab)(m + n + q ) + (bc)(m + n + q ) + (ac)(m + n + q ) = (ab)(m ) + (ab)(n ) + (ab)(q ) + (bc)(m ) + (bc)(n ) + (bc)(q ) + (ac)(m ) + (ac)(n ) + (ac)(q ) Now, the product is simplified: (ab + bc + ac)(m + n + q ) = abm + abn + abq + bcm + bcn + bcq + acm + acn + acq 10. Dividing Polynomials by Monomials To divide a polynomial by a monomial, we simply divide each term of the polynomial by the monomial, and then simplify each quotient to the lowest term. Example: (60x 5 y 4 1x 4 y x 3 y + 4x y ) (6x y) Dividing each term of the polynomial by the monomial can be carried out with the following method: x y 1x y 18x y 4x y (60x 5 y 4 1x 4 y x 3 y + 4x y ) (6x y) = + + 6xy 6xy 6xy 6xy = 10x 3 y 3 x y + 3xy + 4y 11. Dividing Polynomials by Binomials or Polynomials Dividing a polynomial by a monomial or by a polynomial is very similar to the process of dividing a whole number by another whole number. In this process, we divide the first term of the dividend by the first term of the divisor. The result of this division is the first term of the quotient. Then we multiply this result by the divisor, and subtract the product from the dividend. We repeat the same process for the new dividend until we get a remainder equal to zero or smaller than the divisor. All Rights Reserved
6 PERT - A Math Study Guide Let s try an example: (x 5 + 3x 4 + x 3 + 9x + 4x + 4) (x 3 + x + 4) x 3x + 4 x + x + 5 x + 5x + 7x + 14x + 15x x + x +10x 4 3 3x 7x 4x + 15x x + 3x + 15x 3 4x + 4x 0 3 4x + 4x Applying Standard Algorithms or Concepts Standard Algorithm: Many algebra problems can be solved by repeating a particular rule or procedure. For example, the division of a polynomial by another polynomial described above is a sample of standard algorithm. In this example, we repeat dividing the first term of the dividend by the first term of the divisor three times until we get the remainder zero. In fact, we can describe standard algorithm as a set of similar steps to be performed until we reach the answer. As another example, we use the standard algorithm when dividing a polynomial by a monomial. In this case, we divide the first term of the polynomial by the monomial. Then we repeat the same procedure for the following terms of the polynomial. Algorithmic Method in Multiplication: Using standard algorithm is common in multiplying polynomials. As you see in the following example, we repeatedly use the Distributive Property to perform the multiplication of two polynomials: (x + y + z)(ab + bc + ac + abc) = x(ab + bc + ac) + 1. Multiply the first term of the first polynomial by the entire second polynomial. y(ab + bc + ac) +. Multiply the second term of the first polynomial by the entire second polynomial. All Rights Reserved
7 PERT - A Math Study Guide z(ab + bc + ac) 3. Multiply the third term of the first polynomial by the entire second polynomial. You notice that a certain rule is repeated three times to distribute the first polynomial over the second polynomial. Then, the expression obtained in each step can be expanded using the algorithmic method. The expression from the Step 1: x(ab + bc + ac) = xab + xbc + xac 4. x is multiplied by the first term of the second polynomial 5. x is multiplied by the second term of the second polynomial 6. x is multiplied by the first term of the second polynomial Applying Steps 4 6 to the expressions obtained in Steps and 3, we get: 7. y(ab + bc + ac) = yab + ybc + yac 8. z(ab + bc + ac) = zab + zbc + zac Finally, we can put together all the distributions from Steps 4 8, to find the outcome of the product of two polynomials as follows: (x + y + z)(ab + bc + ac + abc) = xab + xbc + xac + yab + ybc + yac + zab + zbc + zac Applying Concepts and Principles: Sometimes when solving a problem we do not have a welldefined algorithm, a formula, or a certain method. But we can put some of these methods and different concepts and principles together to build a solution process for the problem. For example: Assume that D = 180(n ) is the sum of all the interior angles of a regular polygon with n sides. Now, we increase the number of the sides by 7, and want to know that how much the sum of the interior angles in the new polygon is more than the original one. We do not have an algorithm or a particular formula to find such an increase. Rather we can use the concept of the variation in an equation. Let s say D 1 = Sum of the interior angles of the original polygon D = Sum of the interior angles of the new polygon with 7 more sides. Then, D 1 = 180(n ) = 180n 360 Now, we must replace n with (n + 7) All Rights Reserved
8 PERT - A Math Study Guide D = 180[(n + 7) ] = 180(n + 5) = 180n Next, subtract D 1 from D to find the increase D D 1 = (180n + 900) (180n 360) = 180n n = = 160 degrees increase 13. Translate between Lines and Equations As you see some sample points on the coordinate plane below, there are infinite number of such points on a coordinate plane. Each point is identified by two distances: Distance from the horizontal axis or the x axis Distance from the vertical axis or the y axis. For example, the distance of the point A from x axis is 5 and from y axis is 4 These distances are called the coordinates of the point A and are denoted by ( 4, 5) All Rights Reserved
9 PERT - A Math Study Guide You can imagine that each group of these points can be on a straight line. Then this line can be represented by an equation in terms of the coordinates of the points on the line. To find the equation of a line passing through an infinite number of points, we need one of following sets of facts: (a) coordinates of any two points on the line If (x 1, y 1 ) and (x, y ) are the coordinates of two points of a line, then the equation of the line can be obtained using the following formula: y y y y 1 1 y y 1 = x x 1 or y y = x x x x 1 x x 1 (b) coordinates of one point of the line along with the slope of the line If (a, b) is a point on the graph of a line and m is its slope, then the equation of the graph can be obtained using the formula: y b = m(x a) If an equation is given along with a graph of a line on the coordinate plane, we can check the graph against the equation to see whether the equation represents the graph. We substitute only two arbitrary points of the graph in the equation. If the substitution of these points in the equation results in two true numerical equations, then the graph represents the given equation. 14. Simultaneous Linear of Equations in Two Variables All types of systems of two linear equations in two variables can be transferred to the following general model: ax + by = c mx + ny = p When a system of equations is transferred to the above form, then we can use one of the following methods to finds the variables: a. Elimination Method: In this method, we multiply one or both equations by one or two different numbers such that the coefficients of one of the variables in both equations become opposite numbers. Then we add the equations sides by sides. As a result, one of the variables is cancelled out and the simplified equation becomes a one variable equation. We solve this equation in terms one variable. Then we replace the value of this variable in one of the equations, and solve the new equation in terms of the other variable. All Rights Reserved
10 PERT - A Math Study Guide b. Substitution Method: In this method, we solve one of the equations in terms of one variable. Then replace the equivalent of this variable in the second equation. After this replacement, the second equation becomes a one variable equation. We solve this equation in terms of one variable, and replace it in one of the equations. This equation becomes a one variable equation which gives the value of one variable. Replacing the value of this variable in one of the equations generates another one variable equation in terms of the other variable. Solving this equation provides the value of the second variable. This guide was created to help students prepare for the Postsecondary Education Readiness Test. It is an information source and should only be treated as a guide. Although every effort has been made to ensure the accuracy, currency, and reliability of information, users are responsible for making their own assessments of the information within the guide and should seek appropriate professional advice before taking any action based on any information presented here. Any use, distribution, display, or copying of this guide is expressly prohibited. All Rights Reserved
11 P.E.R.T. Math Quiz Directions: Select the correct answer choice for each question. There are 30 questions and you may take as long as you need. Do not use a calculator. After you ve finished, you can use the Answer Key to check your answers. 1. If (4a a + 1) + (5a + 6) = 3, which is the value of a. b. c. d a a + 1 5a + 14? x 8 x. The equation =x 5 is given, where all the terms on the right side are in the form of mixed numbers. Which is the value of x? a. 5 b. 6 c. 9 d. 10 3x 5x 3. If x 17, then which number could be the maximum value of (x + 3x + 5x)? 4 6 a. 48 b. 60 c. 40 d Astronomers discovered that the temperature on the surface of Mars changes according to the inequality 56 t , in degrees Celsius. Which is the range of the temperatures? a. 141 t 9 b. 9 t 141 c. 141 t 56 d. 141 t 56
12 P.E.R.T. Math Quiz 5. Given x 14x + 50 = 1, which is the value of x x 50? a. 194 b. 4 c. 391 d The following equations are given: x + 3x 4 = 0 y 3y + 4 = 0 Which of the following expressions is a real number? a. x + y b. x y c. x d. y 7. The length of a rectangle is 11 feet longer than three times its width. If the perimeter of the rectangle is 38 feet, which is the measure of a width? a. 8 ft b. 7 ft c. 4 ft d. ft 8. The sum of three consecutive odd integers is 369. Which is the product of the first two integers of this set? a b c d
13 P.E.R.T. Math Quiz 9. Given a = 1, b = 1, m =, and n =, which is the numerical value of a. 1 b. 1 c. d. a b a + b + m m n b+m+n? 10. The following expression is given: a 5 + b 5 + a 4 + b 4 + a + b + 1 Replacing which pair of a and b results in the smallest numerical value for the expression? a. a = 1 and b = 1 b. a = and b = c. a = and b = d. a = 1 and b = Which is the factorized form of the expression x + x binomials? a. 1 x + 1 x x x b. 1 x x c. x x d. 1 x x 1 x x + 1 x x + x x in terms of sum of two squared 1. Which is the factorized form of a b ac bc? a. (a b)(a b c) b. (a + b)(a + b c) c. (a b)(a + b c) d. (a + b)(a b c)
14 P.E.R.T. Math Quiz 13. Which is the result of simplifying ab(a + b) bc(b + c) + ac(a c) (a + b)(b + c)(a c)? a. 1 b. 0 c. a + b d. abc + a b ac 14. Which is the equivalent of the expression x(y z) + y(z z) + z(x y) (x y + xy + y z + yz + x z + xz )? a. xyz b. 4xyz c. 6xyz d. 6xyz 15. The sides of a triangle ABC are defined by the following expressions: Which is the perimeter of the triangle? a. 3x 3 6x + 30x + 50 b. 3x 3 + 6x + 30x + 40 c. 3x 3 + 6x + 30x + 50 d. 3x 3 + 6x + 30x 40 AB: 3x 3 6x + 19 BC: 1x + x 10 AC: 9x The following expressions are given: Add the sum of P and Q to twice R. a. a b + 3ab + a c + 3ac b. 3a b + ab + 3a c + ac c. 3a b + ab + a c + 3ac d. 3a b + ab + a c + 3ac P = a b + a b + ab Q = b + c + a c + ac R = a b + ac a c
15 P.E.R.T. Math Quiz 17. Subtract the polynomial N from M. Then subtract the result from P. a. x b. x 5 5 c. 5x d. x 4 15 M = 6x 5 + 5x 4 + 4x 3 + 3x + x + 1 N = x 5 + x 4 + 3x 3 + 4x + 5x + 6 P = 6x 5 + 3x 4 + x 3 x 3x The area of the larger rectangle P is defined by 4x 3 + 4x + 5x 1, and the area of the rectangle M is defined by 4x + x. Which expression describes the area of the rectangle N? P M N a. 4x 3 + 6x 1 b. 4x 3 5x 1 c. 4x 3 + 4x + 1 d. 4x 3 + 4x Which is equivalent to the product of (m n + mn) and (mn mn)? a. m 3 n 3 + m 3 n + m n 3 m n b. m 3 n 3 m 3 n + m n m n c. m 3 n 3 m 3 n + m n 3 m n d. m 3 n 3 m 3 n + m n 3 m n 0. Which is equivalent to the product of (x + 1), (x + 1), and (x 3 + 1)? a. x 6 + x 5 + x 4 + x 3 + x + x + 1 b. x 6 + x 5 + x 4 + x 3 + x + x + 1 c. x 6 + x 5 + x 4 + x 3 + x + x + 1 d. x 6 + x 5 + x 4 + x 3 + x + x + 1
16 P.E.R.T. Math Quiz 1. Which is the quotient of the following division? a. m 3 n b. m n c. m n d. mn (m 5 n 4 + m 4 n 5 + m 3 n 3 + m n) (m 3 n 3 + m n 4 + mn + 1). Which are the quotient and the remainder of the following division? (x 6 + x 5 + x 4 + x 3 + x + x + 1) (x 4 + x 3 + x ) a. Quotient: x + 1; remainder: x + 1 b. Quotient: x 3 + 1; remainder: x + 1 c. Quotient: x 1; remainder: x 1 d. Quotient: x + 1; remainder: x Which is the quotient of dividing (a 7 b 6 c 5 + a 6 b 6 c 6 a 5 b 4 c 5 a 3 b 3 c 3 + a b 3 c 4 ) by a b 3 c 4? a. abcabc abc b. abcabc abc c. abcabc abc d. abcabc abc 4. Which is the quotient of the following division? a. a b + c b. a + b c c. a b c d. a + b + c (a b ac + ab bc + abc c ) (ab c)
17 P.E.R.T. Math Quiz 5. If (x 11)(x + 1) = 0, then which statement is true? a. x = 11 and x = 1 b. x = 11 or x = 1 c. x = 11 and x = 1 d. x = 11 or x = 1 6. If x in the expression (x + 9) 10 is decreased by 6, how much will the value of the expression be changed? a. Will be decrease by 8 b. Will be decreased by c. Will be decreased by 1 d. Will be decreased by 8 7. Which set of graphs represents the graphs of the following system? x y=5 3x y =1 5x + 3y = 1 a. b.
18 P.E.R.T. Math Quiz c. d. 8. Which set of equations represents the following graphs of lines? a. 3x + y = 10 x y = 6 b. 3x + y = 10 x y = 6 c. 3x + y = 10 x y = 6 d. 3x + y = 10 x y=6
19 P.E.R.T. Math Quiz 9. The perimeter of a rectangular land is 40 yards, and its length is 0 yards longer than its width. What are the dimensions of the land? a. a = 50 yards, b = 70 yards b. a = 70 yards, b = 50 yards c. a = 60 yards, b = 80 yards d. a = 10 yards, b = 100 yards (x + y) = 3(x y) 30. Solve x+ 3 = y+3 a. x 5, y b. x 5, y c. x 5, y d. x 5, y This guide was created to help students prepare for the Postsecondary Education Readiness Test. It is an information source and should only be treated as a guide. Although every effort has been made to ensure the accuracy, currency, and reliability of information, users are responsible for making their own assessments of the information within the guide and should seek appropriate professional advice before taking any action based on any information presented here. Any use, distribution, display, or copying of this guide is expressly prohibited.
20 P.E.R.T Math Quiz Answer Key Answer Key P.E.R.T. Math Quiz Questions 1 15 Question Answer b a d a c c d c b d a d b c b Answer Key P.E.R.T. Math Quiz Questions Question Answer d b d d b c a b d d c a c b a Objective: Linear Equations 1. Answer: (b) Solution. (4a a + 1) + (5a + 6) = 3 Given Equation 4a a a + 6 = 3 Remove the parentheses. 11a + 10 = 3 Combine the like terms on the left. 11a = Subtract 10 from each side, and then simplify. a = Divide each side by 11. 4a a + 1 5a + 14 = Substitute a = in the given expression. = Find the products. = 3 Simplify the fraction. Objective: Linear Equations. Answer: (a) Solution. x 8 x =x Given
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 informationFlorida 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 informationFactoring 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 informationVocabulary 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 informationAlgebra 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 informationMATH 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 informationHigher 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 information1.3 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 informationSPECIAL 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 informationMathematics Placement
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.
More informationMTH 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 informationAlgebra 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 informationDefinitions 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 informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationBEGINNING 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 informationPolynomials. 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 informationIV. 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 informationCopy 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 informationMath 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 informationMATH 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 informationAnswer 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 information1.3 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 informationHow 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 informationReview 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 informationMATH 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 informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationMATH 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 informationModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers Basic Math 1.2 - The Number Line Basic Math 1.3 - Addition of Whole Numbers, Part I
ModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers 1) Read whole numbers. 2) Write whole numbers in words. 3) Change whole numbers stated in words into decimal numeral form. 4) Write numerals in
More information1.1 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 informationWhat 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 informationQuick 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 informationFACTORING 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 informationName 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 informationHIBBING 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 informationAnswers 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( ) 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 informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationFACTORING 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 informationMATH 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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationCAHSEE 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 informationBig 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 informationPre-Calculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
More informationAssessment Anchors and Eligible Content
M07.A-N The Number System M07.A-N.1 M07.A-N.1.1 DESCRIPTOR Assessment Anchors and Eligible Content Aligned to the Grade 7 Pennsylvania Core Standards Reporting Category Apply and extend previous understandings
More informationHow To Factor By Gcf In Algebra 1.5
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
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationMathematics. Mathematical Practices
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More information6.1 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 informationChapter 4 -- Decimals
Chapter 4 -- Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789
More informationALGEBRA 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 informationAlgebraic 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 informationStudents 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 informationMATH 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 informationAlgebra 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 informationSUNY 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 informationAlgebra 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 informationFactor 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 informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationof 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 informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationSolving 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 informationMathematics, Basic Math and Algebra
NONRESIDENT TRAINING COURSE Mathematics, Basic Math and Algebra NAVEDTRA 14139 DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. PREFACE About this course: This is a self-study
More informationEQUATIONS 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 informationFactoring 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 informationAlgebra 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 informationHow 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 informationUnderstanding 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 informationCAMI 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 information5.1 FACTORING OUT COMMON FACTORS
C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.
More informationPERT Mathematics Test Review
PERT Mathematics Test Review Prof. Miguel A. Montañez ESL/Math Seminar Math Test? NO!!!!!!! I am not good at Math! I cannot graduate because of Math! I hate Math! Helpful Sites Math Dept Web Site Wolfson
More informationMath 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 informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationAlgebra 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 informationMATH-0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationVocabulary Cards and Word Walls Revised: June 29, 2011
Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationAlgebra I Teacher Notes Expressions, Equations, and Formulas Review
Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts
More informationCommon Core State Standards for Mathematics Accelerated 7th Grade
A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting
More informationAnchorage 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 informationAlgebra 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 informationAlgebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills
McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.
More informationAlgebra 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 informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationFlorida 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 informationNSM100 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 informationCOLLEGE 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 information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More informationUNIT 5 VOCABULARY: POLYNOMIALS
2º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Algebraic Language Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to
More informationA 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 informationMath. MCC6.RP.1 Understand the concept of a ratio and use
MCC6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, The ratio of wings to beaks in the bird house at the zoo was 2:1,
More informationIndiana 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 informationA 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