M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20

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1 Class: Date: ID: A M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which graph represents an odd-degree polynomial function with two x-intercepts?. A. C. B. D. 2. Which of the following statements is FALSE A. A quartic function could have two pairs of equal real zeros B. A cubic function could have just one distinct real zero C. A quintic function must have at least one real zero D. A polynomial function must have at least one real zero 3. Divide: (4x x 21) (x + 7) A. 4x 21 with a remainder of 7 C. 4x + 28 with a remainder of 7 B. 4x 7 with a remainder of 7 D. 4x 4 with a remainder of 7 4. What is the remainder when x 4 8x 3 6x x + 5 is divided by x 2? A. 2 B. 47 C. 1 D. 79 1

2 ID: A 5. The graph of a polynomial function of degree 5 is shown. Which statements are true? i) The function has an even degree. ii) The function has two zeros of multiplicity 2. iii) The equation of the function has a negative leading coefficient. iv) The y-intercept is positive. A. i, ii, iv B. i, ii, iii C. ii, iii, iv D. i, iii, iv 6. Determine the value of k so that x + 5 is a factor of x x x + k. A. k = 55 C. k = 1 B. k = 1 D. k = 55 Numeric Response 1. To the nearest hundredth, determine the solution to the equation: nearest tenth. 8x 11 x 2 = 8. Round your final answer to the 2

3 ID: A Written Response: Make sure you show all relevant work in each of the following questions. 1. Draw a SKETCH of a graph of a polynomial function, P( x) with the following properties: [2] a negative leading coefficient a double at x = 2 a triple at x = 5 a factor of ( x 6) that has a multiplicity of 1 a root at x = 5 such that the total degree of P( x) is 9. 3

4 ID: A 2. The graph of a polynomial function, P( x) has a single root at 6, a single root at 3, and a triple root at 2. P( x) also passes through the point Q(3, 3). Determine the equation of P( x) and write your final answer in factored form [2], 3. Create a rational function written in factored form where: the function has one vertical asymptote at x = 7, a horizontal asymptote at y = 3, and a point of discontinuity at Ê Ë Á8,yˆ [2] BONUS [+1]: Suppose the point of discontinuity was at Ê Ë Á8,2ˆ, what would be the specific equation for this rational function *HINT there is only one possible answer*. 4. For the graph of this rational function, state the domain and write the equations of any asymptotes and the coordinates of any point of discontinuity (if it exists). [2] 2x + 4 y = x 2 + 3x + 2 4

5 ID: A 5. A student is analyzing the graph of the function y = g( x). She correctly deduces that the range of the function Ï y = g( x) is Ô Ì y 0 y 7,y R Ô ÓÔ Ô. She makes four statements about the graph y = g ( x ). Statement 1: The point Ê Ë Á 5,9ˆ lies on the graph of y = g( x) Statement 2: The graph of y = g( x) has no x-intercepts Statement 3: The graph of y = g( x) has no points located in quadrants three and four. Statement 4: The maximum value of y = g( x) is 7. Which of these statements must be true (if any) and which of these statements must be false (if any)? [1] 6. Factor completely the following polynomial function: P( x) = 15x 3 + 7x 2 14x 8. Your work must show the use of the factor theorem and/or synthetic division as needed. SIMPLY STATING THE ALL THE FACTORS WITH NO SUPPORTING ALGEBRAIC WORK - aka finding them on your calculator and filling them in is worth ZERO marks. [2].. 7. For the graph of y = f(x) shown below, what are the domain and range of y = f(x)? [2] 5

6 ID: A M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Answer Section MULTIPLE CHOICE 1. D 2. D 3. D 4. B 5. C 6. D NUMERIC RESPONSE SHORT ANSWER Determine the equation of P( x) and write your final answer in factored form [2 marks] P( x) = 1 18 ( x + 6) ( x + 3) ( x 2) 3 3. f( x) = 3( x 8) ( x + a) ( x 8) ( x + 7) f( x) = 3( x 8) ( x 18) ( x 8) ( x + 7) *THERE are many potential answers for a *THERE are NO other potential answers 1

7 ID: A 4. domain: x 1 and x 2; vertical asymptote: x = 1; hole: ( 2, 2); horizontal asymptote: y = 0 5. Statement 2 must be FALSE, the remaining statements cannot be defined as either true or false with 100% certainty, so NO statements must be true. 6. ( 5x + 4) ( 3x + 2) ( x 1) 7. Domain: 5 x 2 Range: 0 y 2 2

8 Class: Date: ID: B M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which graph represents an odd-degree polynomial function with two x-intercepts?. A. C. B. D. 2. Which of the following statements is FALSE A. A quartic function could have two pairs of equal real zeros B. A polynomial function must have at least one real zero C. A cubic function could have just one distinct real zero D. A quintic function must have at least one real zero 3. Divide: (2x 2 15x + 31) (x 3) A. 2x + 31 with a remainder of 4 C. 2x + 3 with a remainder of 4 B. 2x 9 with a remainder of 4 D. 2x + 6 with a remainder of 4 4. What is the remainder when x 4 + 8x 3 + 3x 2 + 9x + 9 is divided by x + 2? A. 45 B. 77 C. 2 D. 33 1

9 ID: B 5. The graph of a polynomial function of degree 5 is shown. Which statements are true? i) The function has an even degree. ii) The function has two zeros of multiplicity 2. iii) The equation of the function has a negative leading coefficient. iv) The y-intercept is positive. A. i, iii, iv B. i, ii, iv C. i, ii, iii D. ii, iii, iv 6. Determine the value of k so that x + 7 is a factor of x x x + k. A. k = 119 C. k = 1 B. k = 119 D. k = 1 Numeric Response 1. To the nearest hundredth, determine the solution to the equation: nearest tenth. 5x 6 x + 2 = 1. Round your final answer to the 2

10 ID: B Written Response: Make sure you show all relevant work in each of the following questions. 1. Draw a SKETCH of a graph of a polynomial function, P( x) with the following properties: [2] a positive leading coefficient a single at x = 1 a double at x = 5 a factor of ( x 2) that has a multiplicity of 1 a root at x = 1 such that the total degree of P( x) is 7. 3

11 ID: B 2. The graph of a polynomial function, P( x) has a single root at 5, a triple root at 2, and a single root at 1. P( x) also passes through the point Q(6, 224). Determine the equation of P( x) and write your final answer in factored form [2], 3. Create a rational function written in factored form where: the function has one vertical asymptote at x = 0, a horizontal asymptote at y = 4, and a point of discontinuity at Ê Ë Á4,yˆ [2] BONUS [+1]: Suppose the point of discontinuity was at Ê Ë Á4,12ˆ, what would be the specific equation for this rational function *HINT there is only one possible answer*. 4. For the graph of this rational function, state the domain and write the equations of any asymptotes and the coordinates of any point of discontinuity (if it exists). [2] 2x + 8 y = x 2 + 6x + 8 4

12 ID: B 5. A student is analyzing the graph of the function y = g( x). She correctly deduces that the range of the function Ï y = g( x) is Ô Ì y 0 y 2,y R Ô ÓÔ Ô. She makes four statements about the graph y = g ( x ). Statement 1: The point Ê Ë Á0,9ˆ lies on the graph of y = g( x) Statement 2: The graph of y = g( x) has no x-intercepts Statement 3: The graph of y = g( x) has no points located in quadrants three and four. Statement 4: The maximum value of y = g( x) is 2. Which of these statements must be true (if any) and which of these statements must be false (if any)? [1] 6. Factor completely the following polynomial function: P( x) = 6x x 2 + x 10. Your work must show the use of the factor theorem and/or synthetic division as needed. SIMPLY STATING THE ALL THE FACTORS WITH NO SUPPORTING ALGEBRAIC WORK - aka finding them on your calculator and filling them in is worth ZERO marks. [2].. 7. For the graph of y = f(x) shown below, what are the domain and range of y = f(x)? [2] 5

13 ID: B M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Answer Section MULTIPLE CHOICE 1. B 2. B 3. B 4. A 5. D 6. B NUMERIC RESPONSE SHORT ANSWER Determine the equation of P( x) and write your final answer in factored form [2 marks] P( x) = 1 2 ( x 5) ( x 2) 3 ( x + 1) 3. f( x) = 4( x 4) ( x + a) ( x 4) ( x + 0) f( x) = 4( x 4) ( x + 8) ( x 4) ( x + 0) *THERE are many potential answers for a *THERE are NO other potential answers 1

14 ID: B 4. domain: x 2 and x 4; vertical asymptote: x = 2; hole: ( 4, 1); horizontal asymptote: y = 0 5. Statement 2 must be FALSE, the remaining statements cannot be defined as either true or false with 100% certainty, so NO statements must be true. 6. ( 2x + 5) ( 3x 2) ( x + 1) 7. Domain: 4 x 5 Range: 0 y 2 2

15 Class: Date: ID: C M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which graph represents an odd-degree polynomial function with two x-intercepts?. A. C. B. D. 2. Which of the following statements is FALSE A. A polynomial function must have at least one real zero B. A cubic function could have just one distinct real zero C. A quintic function must have at least one real zero D. A quartic function could have two pairs of equal real zeros 3. Divide: (4x 2 26x + 26) (x 5) A. 4x 6 with a remainder of 4 C. 4x + 5 with a remainder of 4 B. 4x + 26 with a remainder of 4 D. 4x + 10 with a remainder of 4 4. What is the remainder when x 4 + 8x 3 + 3x 2 7x + 6 is divided by x + 4? A. 4 B. 14 C. 174 D

16 ID: C 5. The graph of a polynomial function of degree 5 is shown. Which statements are true? i) The function has an even degree. ii) The function has two zeros of multiplicity 2. iii) The equation of the function has a negative leading coefficient. iv) The y-intercept is positive. A. i, iii, iv B. i, ii, iii C. i, ii, iv D. ii, iii, iv 6. Determine the value of k so that x + 2 is a factor of x x x + k. A. k = 1 C. k = 34 B. k = 34 D. k = 1 Numeric Response 1. To the nearest hundredth, determine the solution to the equation: nearest tenth. 9x 10 x + 5 = 1. Round your final answer to the 2

17 ID: C Written Response: Make sure you show all relevant work in each of the following questions. 1. Draw a SKETCH of a graph of a polynomial function, P( x) with the following properties: [2] a positive leading coefficient a single at x = 4 a triple at x = 3 a factor of ( x 2) that has a multiplicity of 2 a root at x = 6 such that the total degree of P( x) is 7. 3

18 ID: C 2. The graph of a polynomial function, P( x) has a double root at 6, a single root at 6, and a double root at 1. P( x) also passes through the point Q(5, 66). Determine the equation of P( x) and write your final answer in factored form [2], 3. Create a rational function written in factored form where: the function has one vertical asymptote at x = 6, a horizontal asymptote at y = 2, and a point of discontinuity at Ê Ë Á7,yˆ [2] BONUS [+1]: Suppose the point of discontinuity was at Ê Ë Á7, 22ˆ, what would be the specific equation for this rational function *HINT there is only one possible answer*. 4. For the graph of this rational function, state the domain and write the equations of any asymptotes and the coordinates of any point of discontinuity (if it exists). [2] 2x + 2 y = x 2 + 4x + 3 4

19 ID: C 5. A student is analyzing the graph of the function y = g( x). She correctly deduces that the range of the function Ï y = g( x) is Ô Ì y 0 y 7,y R Ô ÓÔ Ô. She makes four statements about the graph y = g ( x ). Statement 1: The point Ê Ë Á9,7ˆ lies on the graph of y = g( x) Statement 2: The graph of y = g( x) has no x-intercepts Statement 3: The graph of y = g( x) has no points located in quadrants three and four. Statement 4: The maximum value of y = g( x) is 7. Which of these statements must be true (if any) and which of these statements must be false (if any)? [1] 6. Factor completely the following polynomial function: P( x) = 10x x 2 31x Your work must show the use of the factor theorem and/or synthetic division as needed. SIMPLY STATING THE ALL THE FACTORS WITH NO SUPPORTING ALGEBRAIC WORK - aka finding them on your calculator and filling them in is worth ZERO marks. [2].. 7. For the graph of y = f(x) shown below, what are the domain and range of y = f(x)? [2] 5

20 ID: C M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Answer Section MULTIPLE CHOICE 1. A 2. A 3. A 4. C 5. D 6. C NUMERIC RESPONSE SHORT ANSWER Determine the equation of P( x) and write your final answer in factored form [2 marks] P( x) = 1 6 ( x 6) 2 ( x + 6) ( x + 1) 2 3. f( x) = 2( x 7) ( x + a) ( x 7) ( x 6) f( x) = 2( x 7) ( x 18) ( x 7) ( x 6) *THERE are many potential answers for a *THERE are NO other potential answers 1

21 ID: C 4. domain: x 3 and x 1; vertical asymptote: x = 3; hole: ( 1,1); horizontal asymptote: y = 0 5. Statement 2 must be FALSE, the remaining statements cannot be defined as either true or false with 100% certainty, so NO statements must be true. 6. ( 5x 2) ( 2x + 5) ( x 1) 7. Domain: x 4 or x 5 Range: y R 2

Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following