( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: Factor completely: 3. Factor completely:

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1 Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. ( + 6)( 6 ). Factor completely: 6a ( a+ )( a a+ ) 6. Factor completely: 0 y. ( y)( + y+ y ) 7. Factor completely: 0 y+ y. ( y)( y) 8. Factor completely: + 9. ( 7) 9. Factor completely: ( y) ( y) 8. ( 6y+ )( 6y 7)

2 0. Factor completely: 8y+ 6y z. ( yz)( y+ z). Write the following quadratic equation in standard form. Identify the quadratic term, the linear term, and the constant term. ( ) =.7 + 0= 0 Quadratic term: Linear term: Constant term: 0. Solve: = 0.7, 0. Solve: = 0.7 {, }. Solve: =.7 { 6 }. Solve: = 7.7 7, 6. Solve: ( + )( ) = ( + )( ).7 {, } 7. Find two consecutive even integers, the sum of whose squares is 0..8 and ; - and - 8. The length of a rectangle is twice the width. If the width is increased by m and the length is increased by m, then this larger rectangle would have an area of 0 sq m. What are the dimensions of the original rectangle?.8 6m m 9. One leg of a right triangle is 7 in. shorter than the other leg. The hypotenuse is in. long. What are the lengths of the legs of the triangle?.8 in. and in.

3 0. Little Adam is sitting in his high chair with a bowl of oatmeal in front of him, ft above the floor. Instead of eating it, he drops it on the floor. The equation that gives the bowl s height above the floor is h = 6t +. When does the bowl hit the ground?.8 After sec. A photograph and its frame cover sq in. The frame is in. wide. The length of the photograph is in. longer than twice the width. What are the dimensions of the photograph?.8 in. 7 in.. 6 Evaluate:. 6. Evaluate: Evaluate: ( 8 0. ). Evaluate: (. ) 8 6. Evaluate: Evaluate: ( ) Evaluate: ( 6 ) Evaluate:. 6 6

4 0. Evaluate: Simplify. Write with only positive eponents. ( y) ( 8 y ). y 9 7. Simplify. Write with only positive eponents. a b. a 8b 6. Simplify. Write with only positive eponents. 8 y 6y. 6 y 6 8. Simplify. Write with only positive eponents. 6 y y 0 7. y 8 6. Simplify. Write with only positive eponents. y. 9 8 y 6. Reduce to lowest terms. 7ab ab. ab 7. Reduce to lowest terms. 8+.

5 8. Reduce to lowest terms. y 8 + 0y8y. y+ ( + y) 9. Reduce to lowest terms. 8a + b 8a + 6ab+ b. a ab+ b a+ b 0. Reduce to lowest terms. a + a bab b b baa. ( a+ b)( a+ b) b+ a. Perform the indicated operation and reduce to lowest terms. 7 6 y ay y 7ab 9a z ab. 6 9a z. Perform the indicated operation and reduce to lowest terms ( ) ( + ). Perform the indicated operation and reduce to lowest terms. a 8b ab+ 6ab + ab 6b baa a 9b. a b ab. Perform the indicated operation and simplify if possible. y + y y y + y y. y +

6 . Perform the indicated operation and simplify if possible. 7 y y + y. y + y 6. Perform the indicated operation and simplify if possible ( + ) ( + ) 7. Perform the indicated operation and simplify if possible ( )( + )( 8) 8. Simplify the following fraction: Simplify the following fraction: Simplify the following fraction: ( + )

7 . Simplify the following fraction: ( ). Simplify the following fraction: y + y. ( y )( y+ ) y. Perform the indicated division: 8ab 6ab + ab 0ab.6 a 6b + b a. Perform the indicated division: Perform the indicated division: Solve for. + = 8.8 { } 7. Find the solution to the following problem. The sum of the reciprocals of two consecutive even integers is. What are the integers? 8.9 and

8 8. Find the solution to the following problem min Kristin can shell a quart of pecans twice as fast as her brother Brian. Working together they can shell a quart of pecans in 0 min. How long would it take Brian to shell a quart of pecans by himself? 9. Find the solution to the following problem..9 Up mph, down 0 mph Hermes can travel mi up Mt. Olympus at a rate that is 8 mi/hr slower than he rides Pegasus down the same distance. The entire trip takes him min. How fast can he go up? How fast can he ride down? (Hint: Change min to hours.) 60. Evaluate (if possible): Evaluate (if possible): Evaluate (if possible): Evaluate (if possible): ( 6). Not a real number 6. Evaluate (if possible): ( ).

9 6. Evaluate (if possible): Evaluate (if possible):. 67. Evaluate (if possible): Evaluate (if possible): ( ). Not a real number 69. Evaluate (if possible): Evaluate (if possible): Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers

10 7. Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers. 8 y y. 7 y Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers. 8 y. 0 y

11 77. Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers. 8 9 y y. 6 7 y 78. Simplify. (Write the answer with only positive eponents.) Assume that all of the variables represent positive real numbers. ( ) Evaluate, if possible Evaluate, if possible.. Not a real number 8. Evaluate, if possible.. 8. Evaluate, if possible.. 8. Evaluate, if possible. 8 7.

12 8. Evaluate, if possible Evaluate, if possible Convert the following to radicals. a. a 87. Convert the following to radicals. ( ab ). 6ab 8 or ab ab 88. Convert the following to radicals. ( + y ). + y 89. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers.. 9. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers. a. a

13 9. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers. ( y ). y 9. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers. ( y ). ( y ) 9. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers. ( a b) +. a+ b 96. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers. y y. y 97. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers..

14 98. Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers Convert the following radicals to epressions with rational eponents and simplify where possible. Assume that all variables represent positive numbers Simplify. Assume that the variables represent positive real numbers. 6 6 y 8. 6 y 0. Simplify. Assume that the variables represent positive real numbers. 7 y 6. y 0. Simplify. Assume that the variables represent positive real numbers. 6 8 y. y 0. Simplify. Assume that the variables represent positive real numbers. y + 8y+ 6. y + 0. Simplify Simplify

15 06. Simplify Simplify Simplify Simplify Simplify. Assume that all variables represent positive numbers. 6.. Simplify. Assume that all variables represent positive numbers. y 9. y y. Simplify. Assume that all variables represent positive numbers. 8 y z 8. yz z. Simplify. Assume that all variables represent positive numbers. y. y y. Simplify. Assume that all variables represent positive numbers. 6 y. 6y y

16 . Simplify. Assume that all variables represent positive numbers. 8y. y y 6. Simplify. Assume that all variables represent positive numbers. y. y y 7. Simplify. Assume that all variables represent positive numbers Simplify. Assume that all variables represent positive numbers. 6 9 y. y y 9. Perform the indicated operation Perform the indicated operations. Assume that all variables represent positive numbers. 9 y + y. 0 y. Perform the indicated operations Perform the indicated operations

17 . Perform the indicated operations. Assume that all variables represent positive numbers. p + p p 6p. 7p p p or ( 7p ) p. Perform the indicated operations. Assume that all variables represent positive numbers. 6 0 y 7y y + y 0 y. 9 y + y 0y. Perform the indicated operations.. 6. Perform the indicated operations Perform the indicated operations. Assume that all variables represent positive numbers ( ) or 8. Perform the indicated operations. Assume that all variables represent positive numbers Perform the indicated multiplication and simplify the answer. ( + 0 ) Perform the indicated multiplication and simplify the answer. Assume that all variables represent positive numbers. + ( ). +

18 . Perform the indicated multiplication and simplify the answer. ( + )( 7 ) Perform the indicated multiplication and simplify the answer. Assume that all variables represent positive numbers. y + y ( )( ) y y. Perform the indicated multiplication and simplify the answer. ( 6 ). 8. Perform the indicated multiplication and simplify the answer. Assume that all variables represent positive numbers. ( y + ). y+ y +. Perform the indicated multiplication and simplify the answer. ( 7 + )( 7 ). 6. Perform the indicated multiplication and simplify the answer. Assume that all variables represent positive numbers. + ( )( ). 7. Rationalize the denominator of the following radical epressions. +. 6

19 8. Rationalize the denominator of the following radical epressions Rationalize the denominator of the following radical epressions. Assume that all variables represent positive numbers. y 6 6 8y y y 0y 9 8y 0. Find the solution(s) of the following radical equation. + = 0.6 { }. Find the solution(s) of the following radical equation. 6 = +.6 { }. Find the solution(s) of the following radical equation =.6 { }. Find the solution(s) of the following radical equation. + + =.6 { }. Find the solution(s) of the following radical equation. + =.6,. Epress the following in the form a+ bi and simplify. 6.7 i

20 6. Epress the following in the form a+ bi and simplify i 7. Epress the following in the form a+ bi and simplify Epress the following in the form a+ bi and simplify Epress the following in the form a+ bi and simplify i 0. Epress the following in the form a+ bi and simplify Epress the following in the form a+ bi and simplify i. Perform the indicated operations. Epress the answer in the form a+ bi. + i + ( + i ).7 + i. Perform the indicated operations. Epress the answer in the form a+ bi. i + 7i ( ) ( ).7 i

21 . Perform the indicated operations. Epress the answer in the form a+ bi. 6+ 8i ( ) i. Perform the indicated operations. Epress the answer in the form a+ bi. i i ( ).7 + i 6. Perform the indicated operations. Epress the answer in the form a+ bi. + 6i + 9i ( )( ) i 7. Perform the indicated operations. Epress the answer in the form a+ bi. 6 i 6+ i ( )( ) Perform the indicated operations. Epress the answer in the form a+ bi. ( + 7i ).7 + 8i 9. Perform the indicated divisions. Epress the answer in the form a+ bi. + i i.7 i 60. Perform the indicated divisions. Epress the answer in the form a+ bi. + i.7 6 i 6. Perform the indicated divisions. Epress the answer in the form a+ bi. i i.7 + i

22 6. Perform the indicated divisions. Epress the answer in the form a+ bi. i i.7 + 6i 6. Simplify..7 i 0 6. Simplify..7 i i 6. Simplify..7 i i 66. Simplify..7 i i 67. Graph the following equation: y = Graph the following equation: y = + 6.

23 69. Graph the following equation: y = Find the domain and range, and determine if the following relation is a function. + y = 6 6. Domain = {All real numbers} Range = {All real numbers} function 7. Find the domain and range, and determine if the following relation is a function. y = + 6. Domain = {All real numbers} yy Range = { } function 7. Find the domain and range, and determine if the following relation is a function. y = 7. If g ( ) g( ) =, find the following: 6. Domain = { } Range = { yy 0 } function If ( ) ( ) f = + 7, find the following: [ ] f If ( ) f = + 7 following: f g ( ) ( ) and g ( ) =, find the 6. 68

24 76. If ( ) f = + 7 following: f + g ( ) ( ) and g ( ) =, find the If ( ) f = + 7 following: f g ( ) ( ) and g ( ) =, find the If g ( ) ( ) + gh ( ) ga =, find the following: 6. a+ h 79. If g ( ) ga ( + h) =, find the following: 6. a+ h 80. If g ( ) =, find the following: ( + ) ( ) ga h ga h If ( ) f = + 7 following: f g ( ( )) and g ( ) =, find the Sketch the graph of the following linear equation and find all intercepts. y = 6. int: ( 0, ) y int: ( 0, )

25 8. Sketch the graph of the following linear equation and find all intercepts. = 6. int: ( 0, ) 8. Sketch the graph of the following linear equation and find all intercepts. y = 6. y int: ( 0, ) 8. Find the slope of the line passing through the following points: 76, (, ) and ( ) Find the slope of the line passing through the following points: 7, (, ) and ( ) 6. Undefined 87. Find the slope of the line passing through the following points:, (, ) and ( ) 6. 0

26 88. Find the slope of the following line. y = Find the slope of the following line. = 6. Undefined 90. Find the slope of the following line. y = Find the equation of the following line. Write the answer in slope-intercept form, if possible. m =, through ( 07, ) 6. y = Find the equation of the following line. Write the answer in slope-intercept form, if possible., m is undefined, through ( ) 6. = 9. Find the equation of the following line. Write the answer in slope-intercept form, if possible. 76, through (, ) and ( ) 6. 9 y = 9. Find the equation of the following line. Write the answer in slope-intercept form, if possible. 0, parallel to + y = through ( ) 6. y = + 9. Find the equation of the following line. Write the answer in slope-intercept form, if possible., perpendicular to y = 0 through ( ) 6. y = Find the equation of the following line. Write the answer in slope-intercept form, if possible., m =, through ( ) 6. y =

27 97. Find the equation of the following line. Write the answer in slope-intercept form, if possible., m = 0, through ( ) 6. y = 98. Find the equation of the following line. Write the answer in slope-intercept form, if possible. through (,6) parallel to y + = 6. y = Find the equation of the following line. Write the answer in slope-intercept form, if possible. through (,6) perpendicular to y + = 6. = 00. Sketch the graph of the solution set of: 6.7 y 0. Sketch the graph of the solution set of: + y > 6.7

28 0. Sketch the graph of the solution set of: y 6 y (a) The area of an equilateral triangle varies directly as the square of the side. Write an equation for this sentence. (b) The area is 9 sq in. when the side is 6 in. long. Determine the constant of variation. 6.8 (a) A = ks (b) k = 0. The weight of a body near the surface of the earth varies inversely as the square of the distance of that body from the center of the earth. A man standing on the surface of the earth is approimately 000 mi from the center of the earth. If he weighs 00 lb there, how much would he weigh if he were 00 mi above the surface of the earth? 6.8 Approimately 6 lb 0. y varies jointly as and z. y= when = and z=. What is the value of y when =8. and z=0? Use the Etraction of Roots Theorem to find the solutions of: + 8 = 0 7. ±9i 07. Find the solutions of the following quadratic equation by completing the square = or 08. If the square is completed on the equation 8 + = 0, what is a result? 7. ( ) =

29 09. Use the Quadratic Formula to find the solution of the following equation = 0 7. ± i 0. Use the discriminant to characterize the solutions of the following quadratic equation. 9 + = 0 7. one rational solution. Use the discriminant to characterize the solutions of the following quadratic equation. 7 = 0 7. two rational solutions. Use the discriminant to characterize the solutions of the following quadratic equation. + = 0 7. two irrational solutions. Use the discriminant to characterize the solutions of the following quadratic equation. + 6= 0 7. two imaginary solutions. Determine the value(s) of m so that the following quadratic equation will have one rational solution. + m = 0 7. m =. Determine the real values of m so that the following quadratic equation will have two imaginary solutions. m + + = 0 7. m > 6. Find the solution of the following problem. If the solution is irrational, determine a decimal approimation of the solution. One pipe can fill a tank hr faster than another pipe. Together they fill the tank in hr. How long does it take each pipe to fill the tank? hr and hr

30 7. Find the solution of the following equation. 7 + = ±, ± 8. Find the solution of the following equation. = Find the solution of the following equation. ( ) ( ) + = , 0. Find the solution of the following equation. + 8= ,. Find the solution of the following equation. 6 + = , 7. Find the solution of the following equation. 0 + = ,. Find the solution of the following equation. + = ±,. Using graphs, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. + y = 6 y = 9. {( ) }, ; independent

31 . Using graphs, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. y = y = 0 9. ; inconsistent 6. Using graphs, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. 6 y = 0 y = 9. {( y, ) y } = ; dependent 7. Using the substitution method, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. + y = + y = 9., ; independent 8. Using the substitution method, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. 6 y = 0 9 y = 9. ( ) { y, 6 y 0 } dependent = ;

32 9. Using the elimination method, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. + 8y = 7 + y = 9. ; inconsistent 0. Using the elimination method, determine the solution of the following linear system of equations. Identify the system as independent, inconsistent, or dependent. y = + 6y = , ; independent. Using the elimination method, determine whether the following linear system is independent, inconsistent, or dependent. If the system is independent, find the solution. + y+ z = + y z = + y+ z = 9. ( 6,, 8) ; independent