Common Core Algebra 2 Review Session 1

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1 Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x (4) 12x 4 +16x When the expression 5x( 2x + 5) x 2 + 4x is simplified it is equivalent to (1) 8x x (2) 6x x (3) 9x x (4) 9x x 3. For x = 6, the rational expression x2 4x 2x + 4 has a value of (1) 7 5 (2) 15 2 (3) 8 3 (4) The solution to x = 7 is which of the following? 3 (1) 11 2 (2) 49 6 (3) 40 3 (4) For which of the following values of k will the equation 5( 2x + 3) 8 = 10x + k be an identity? (1) k = 6 (2) k = 5 (3) k = 7 (4) k = Which of the following equations has no values of x in its solution set? (1) 8x + 3 = 6x 7 (2) 4( x +1) 2x = x + 5 (3) 3( 2x +1) 7 = 6x 4 (4) 5( x + 2) = 2( 2x 3) + x

2 7. If y = 10 2 x +11, then what is the value of y when x = 25? (1) 2 (2) 48 (3) 22 (4) The exponential expression ( 2x ) 2 3 ( 5x ) 4 can be rewritten equivalently as (1) 40x 10 (2) 10x 10 (3) 30x 24 (4) 24x The monomial 36x 4 y 10 can be written as the product of 3x 2 y 2 with which of the following (1) 12x 2 y 5 (2) 12x 2 y 8 (3) 12x 2 y 8 (4) 12x 2 y Which of the following values of x is a zero of x 3 + 5x 2 +11x +15 = 0? (1) x = 5 (2) x = 3 (3) x = 7 (4) x = If the product of the binomial ( 2x 5) with the trinomial ( 3x 2 + 2x 5) is formed, what is the coefficient of the x 2 term? (1) 11 (2) 2 (3) 7 (4) Write the following expression as the product of two binomial factors. ( x + 6) ( x + 5) + ( x + 6) ( x 2) Factor each expression as completely as possible: 13) x 2 +13x ) 6x 2 17x 14 15) 10x 2 19x ) x 4 + 6x ) 10x 6 +13x ) x 4 10x 2 + 9

3 Still factoring: Hint: Do not combine like terms before attempting to factor! 19) x 3 + 3x 2 + 2x + 5x 2 +15x ) 8x 3 +10x 2 14x 12x 2 15x ) 8x x x + 20x x ) x 4 4x 2 + 8x 3 32x +12x ) ( 4x 2 + 5x) 2 5( 4x 2 + 5x) 6 24) If the function y = 9 2x 3 were placed in the form y = axb, where a and b are real numbers, then what is the value of a + b? (1) 3 2 (2) 6 (3) 15 (4) ) Which of the following best describes the end behavior of the polynomial y = 3x 4 + 5x 2 + 3x 1? (1) y = x 4 (2) y = 3x 4 (3) y = 3x (4) y = 5x 2

4 26) Which of the following could be the equation of the polynomial graph shown below? (1) y = x( x + 5) ( x 3) (2) y = x( x + 3) ( x 5) (3) y = x 2 ( x + 5) ( x 3) (4) y = x 2 ( x + 3) ( x 5) 27) The cubic polynomial shown graphed below passes through the point ( 6,192) as shown. Which of the following is the value of the leading coefficient of the polynomial? (1) 7 (2) 3 (3) 1 3 (4) 5 28) If a, b, and c all represent positive numbers with a < b < c and f x the following describes all values of x for which f x ( ) 0? ( ) = ( x + a) ( x b) 2 ( x c) then which of (1) { x a x b} (2) { x x < a or x c} (3) { x b < x < b} (4) { x a x c} 29) Which of the following values of x is not in the domain of the function y = (1) x = 9 (2) x = 2 (3) x = 14 (4) x = 2 30) If f ( x) = x x +1, then f 1 x ( ) equals x 9 x 2 5x 14? (1) x +1 x (2) x x +1 (3) x 1 x (4) x 1 x

5 31) The rational expression x2 + 4x 21 can be simplified to which of the following, assuming x ±7? x 2 49 (1) 3 (2) x + 7 4x 21 (3) (4) x 3 7 x 7 49 x 7 32) When the square of a number is decreased by nine and then divided by six less than twice the same number, the result is (1) three more than twice the number (2) one third of two less than the number (3) one half of three more than the number (4) four less than one fourth of the number 33) Given x 0 and y 0, the expression 8x2 y 5 3x 4 16 ( x 3 y) 2 is equivalent to 9y 4 (1) 2 5 x4 y 6 (2) 18x 6 y 3 (3) 3 2 x 8 y 7 (4) 20 3 x 2 y 5 34) Which of the following represents the sum 1 x + 1 y expressed as a single fraction? (1) 2 x + y (2) x + y xy (3) 1 x + y (4) xy x + y 35) Which of the following is equivalent to the difference x 2 16 x 2x x? 2 (1) x + 2 x + 4 (2) x + 4 x + 8 (3) x 6 x + 2 (4) x 3 x ) For x 0, the complex fraction 1 1 x simplifies to 1+ 1 x (1) 1 (2) x2 1 x (3) (4) x 1 x

6 37) The complex fraction 1 x + y 1 x y can be rewritten as (1) 1 x 2 + xy (2) 1 y 2 (3) y x 2 + y 2 (4) 1 y y 2 + xy 38) The quotient x 10 x + 2 can be written as (1) 5 (2) 1 12 x + 2 (3) 1 5x (4) 1 8 x ) The equivalent form of 6x 1 3x + 7 when written in the form 2 + b has b equal to: 3x + 7 (1) 1 (2) 7 (3) 12 (4) 15 40) Which of the following is a factor of the cubic polynomial x 3 10x 2 +11x + 70? (1) x +10 (2) x 2 (3) x 7 (4) x ) What is the remainder when x 2 + 8x 7 is divided by x 2? (1) 13 (2) 27 3) 19 (4) 5 42) The smallest solution to the equation 2x x 2 = 30 x + 4 (1) x = 4 (2) x = 6 (3) x = 5 (4) x = 10 is which of the following?

7 43) Which of the following represents the solution set of the equation: 1 16 x 2 = 6 x? (1) { 4, 6} 2) { 8, 2} (3) { 4, 4} (4) { 2, 8} 44) Which of the following values of x lies in the solution set of the inequality x 10 x 5 < 0? (1) x = 0 (2) x = 7 (3) x = 2 (4) x = 15 45) The solution set to the inequality 2x 9 x 1 0 is which of the following? (1) { x 1< x 4.5} (2) { x 1 x 4.5} (3) { x x < 1 or x 4.5} (4) { x x 1 or x 4.5} 46) Given the cubic polynomial f ( x) = x 3 5x 2 4x + 20 a) Find the x-intercepts of this function algebraically. b) Explain why the graph below could not represent that of f ( x).

8 and passes through the point ( 2, 36) as shown. Algebraically determine its equation in factored form. Show how you arrived at your answer. 47) The cubic polynomial below has zeroes only at x = 4 and x = 6 48) Is the following equation an example of an identity? Justify your answer. x 3 + y 3 = ( x + y) ( x 2 xy + y ) 2 49) Determine any value(s) of x that do not lie in the domain of the function f ( x) = x2 2x 15 2x 2 19x Explain how you arrived at your value(s). 50) For the function f ( x) = x + 8 2x 1, find f 1 1 ( ) Show how you arrived at your answer.

9 51) Simplify the following rational expression for all values for which it is defined. ( x + 5) ( 3x 7) ( x + 5) ( x +1) x 2 x 12 52) Express the following in simplest form. 5x x 36 x 2 10x 2 x 2 4x 12 53) Express the sum as a single fraction in simplest form. x 6 x 2 + 3x x 2 + 4x 54) Write the following complex fraction in simplest form x x 5 5 x

10 55) Rewrite the rational expression 4x3 2x 2 + 8x +10 x 5 polynomial and k is a constant. in the form p( x) + k x 5, where p ( x ) is a quadratic b) Is x 5 a factor of 4x 3 2x 2 + 8x +10? Explain your reasoning. ( ) = x 3 x 2 44x 96 : ( ) = 0, what binomial must be a factor of this polynomial? 56) Given the cubic polynomial f x a) If f 3 b) Given your answer to (a), algebraically determine the two other factors of f ( x). 57) Solve the following equation for all value(s) of x: 5 x + x +1 x 2 = x + 4 x 2 2x

11 58) a) Solve the square root equation x 2 = 13 4x b) Why did an extraneous root get introduced? 59) Which of the following sets of ordered pairs is not be a function? (1) ( 4, 1), ( 1, 7), ( 3, 8), ( 5, 3) (3) ( 2, 8), ( 4,10), ( 6, 8), 8,10 { } { ( )} {( ), ( 6,1), ( 2,10), ( 6, 1) } (4) {( 3, 5), ( 3, 5), ( 6, 7), ( 6, 7) } (2) 2, 5 60) In the following graph, the height of an object, in feet, is given as a function of time in seconds. Which of the following is the range of this function? (1) 0, 5 (3) 176, 576 (2) 0, 11 (4) 0, ) If f x ( ) = 1 2 x + 6, then which of the following values solves the equation f ( x) = 10? (1) 1 (2) 4 (3) 8 (4) 11 62) Which of the following values of x is not in the domain of the function f x ( ) = x 7 2x + 5? (1) 7 (2) 2.5 (3) 5 (4) 7

12 63) In which of the following four graphs is the output not a function of the input? (1) (2) (3) (4) 64) The function f is defined by the formula f x below. Which of the following is the value of f g 2 (1) 18 (2) 14 (3) 5 (4) 9 ( ) = x and the function g is defined by the graph shown ( ) ( )? 65) Given the function f ( x) shown in the graph below, for which of the following intervals is f ( x) > 0? (1) ( 0, 8) (2) 0, 8 (3) ( 4, 6) (4) 4, 6 66) If the point 4, 2 of y = f 1 (1) ( 2, 4) ( ) lies on the graph of y = f ( x), then which of the following points must lie on the graph ( x)? (2) 1 4, 1 2 (3) ( 4, 2) (4) ( 4, 2) 67) Given the function shown below, over which of the following intervals is the function always increasing? (1) 0 < x < 5 (2) 5 < x < 2 (3) 1< x < 4 (4) 9 < x < 5

13 68) Which of the following graphs represents a one-to-one function? (1) (3) (2) (4) 69) Which of the following is the y-intercept of the piecewise defined function g( x) = (1) 5 (2) 6 (3) 1 (4) 8 6x + 5 x < 2 ( x 3) 1 x 2? 70) Given the function y = f ( x) shown graphed below, answer the following questions. a) State the value of f ( 2). b) How many values solve the equation f ( x) = 5? Explain how you arrived at your answer. c) On the interval 0 < x < 4 is the function increasing or decreasing? d) Given g( x) = 2x 7 3 find the value of g( f ( 1) )? e) Is the inverse of f ( x) a function? Explain your answer.

14 71) Find the domain of. 72) Is the function y = x one-to-one? Explain your answer. 73) Given the function y = f x a) Graph the function s inverse, f 1 x b) State the range of f 1 ( x). ( ) shown below do the following. ( ). c) What is the value of f 1 ( 3)? 74) Which of the following is the solution to 2( x 3) 2 = 32? (1) x = 1 and x = 7 (2) x = i and x = 7i (3) x = 3± 4i (4) x = 2 and x = 5 75) Which of the following is the correct simplification of 98? (1) (2) (3) (4) 76) Which of the following is equivalent to ai 7 + bi 16 ci 21, where a, b, and c are real numbers and i = 1? (1) a + ( b c)i (2) b ( a + c)i (3) ( a + c)i b (4) a b + ci 77) Which of the following is the sum of 5 3i and 7 + 8i? (1) 35 24i (2) 12 11i (3) 2 + 5i (4) 2 11i

15 78) If x = 3+ 7i and y = 6 2i then which of the following is equivalent to the expression 2x 4y? (1) i (2) 6 + 8i (3) i (4) 12 7i 79) The product of 6 8i and which of the following will result in a purely imaginary number? (1) 6 + 8i (2) 4 + 3i (3) 3+ 4i (4) 8 + 6i 80) If the product of a 2bi and 2a + 5bi is found, where a and b are real numbers, then the real component of the result is given by (1) 10b 2 2a 2 (2) 9ab (3) 2a 2 +10b 2 (4) 12a 2 b 2 81) The function f x graphs could represent f x ( ) is quadratic and the solutions to f ( x) = 0 are x = 2 ± 3i. Which of the following ( )? (1) (3) (2) (4)

16 82) For which of the following values of b will the equation 4x 2 + bx + 7 = 0 have real solutions? (1) b = 5 (2) b = 9 (3) b = 8 (4) b = 11 83) Explain why the product of 10 4i and 5+ 2i produces a purely real number as a result. 84) Find the solutions to x = 0 in simplest form. 85) Find all solutions to the following equation in simplest a + bi form. ( ) = 3( x + 8) 4 x x ) Evaluate the following complex arithmetic problem. Show all steps and express your final answer in simplest a + bi form. ( 5+ i) ( 2 7i) + ( 8 + 3i) ( 1+ 2i)

17 87) Consider the function f x a) Solve the equation f x ( ) = 4x 2 4x + 5. ( ) = 0 for all solutions, real or complex. b) What does your solution set from (a) tell you about the graph of f ( x)? Explain. 88) Given the quadratic equation ax 2 +10x + 2 = 0, determine all values of a that will result in this equation having non-real solutions. Show the work that leads to your answer.

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power. MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world