P.E.R.T. Math Study Guide

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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

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