Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

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1 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 SE: Express the following polynomial function in factored form: P( x) = 15x x 3 29x 2 11x If ( 2x + 3) is a factor of P( x) = 10x x 2 + kx 6, determine the value of k A list of five functions is given below: 4 Which of the functions shown above represents a polynomial function? 1

2 5 Which of the following is a factor of f( x) = 4x 4 x 3 8x 2 40? a) ( x + 2) b) ( x 4) c) ( x 6) d) ( x + 8) The graph of the polynomial function, f( x) is shown below: 6 What is the minimum possible degree for the polynomial function above and determine an equation of the function in factored form 7 What is the remainder when x 4 7x 3 2x 2 + 2x + 2 is divided by x 2? a) 10 b) 74 c) 42 d) 2 8 P( x) = ax 3 + bx 2 + cx + d, a 1, is an integral polynomial function with 3 and 6 as its roots The graph of y = P( x) is shown below The maximum positive y-intercept is a) 18 b) 3 c) 54 d) 6 2

3 A box with no lid is made by cutting four squares of side length x from each corner of a 10 cm by 20 cm rectangular sheet of metal 9 Using the information given above, answer the following: Find an expression that represents the volume of the box Plot the graph of the function on your calculator and state any restrictions on the value of x Find the value of x, to the nearest hundredth of a centimetre, that generates the maximum possible volume for this box What is the maximum volume of this box, rounded to the nearest cm 3 The graphs of four polynomial functions are shown below: 10 Match three of the graphs numbered above with a statement below that best describes the function The graph that has a positive leading coefficient is graph number The graph of a function that has two different zeros, each with multiplicity 2, is graph number The graph that could be a degree 4 function is graph number 3

4 11 A polynomial function P of the form P(x) = x(x a)(x b) 2 (x c) 3, where a, b, and c are integers, is of degree a) 4 b) 6 c) 7 d) 5 The graph of the function, y = f( x), is shown below: 12 Sketch the graph of y = f( x) and state the domain and range of this function 13 State the coordinates of any invariant points when f( x) = 1 x 3 is transformed to y = f x 2 ( ) 4

5 14 Determine the x-intercept of y = 2 x , to the nearest hundredth, and explain its relationship to the zero of the function 15 Sketch the graph of the following functions and determine the following characteristics for each function below: domain, x- and y-intercepts, equation of vertical asymptotes y = 3x x 2 + 2x 8 y = x + 3 x SE: For the graph of y = 3x + 7, determine the equation of the horizontal asymptote and the range 2x SE: Determine the coordinates of the point of discontinuity on the graph of f( x) = 2x 2 15x + 7 x 7 5

6 ax The graph of the function below can be expressed in the form y = x 2 + bx + c 18 SE: Determine the values of a, b, and c 19 The graph of which function below has 1 vertical asymptote and no horizontal asymptote? a) y = x 4 x b) y = x 2 + 6x + 8 x + 2 c) y = x 2 x 5 d) y = x + 5 x What is the solution of this rational equation, to the nearest tenth if necessary? 10 x 8 = 6 a) x =Ö 63 b) x =Ö 63 c) x =Ö 97 d) x =Ö 97 6

7 Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations Answer Section 1 For P( x) = 10x 3 + x 2 52x + 20, use a guess-and-check method or your calculator to determine a possible factor for P( x) P( 2) = 10( 2) 3 + ( 2) 2 52( 2) + 20 = 0, so ( x 2) is a factor and 2 is a value can be used in synthetic division to further break down the cubic into a quadradic Ê So, P( x) = ( x 2) 10x 2 ˆ + 21x 10 Á = ( x 2) ( 2x + 5) ( 5x 2) 2 For P( x) = 15x x 3 29x 2 11x + 6, use a guess-and-check method or your calculator to determine a possible factor for P( x) P( 1) = 15( 1) ( 1) 3 29( 1) 2 11( 1) + 6 = 0, so ( x 1) is a factor and 1 is a value can be used in synthetic division to further break down the quartic into a cubic Ê So, P( x) = ( x 1) 15x x 2 ˆ + 5x 6 Á Now, use a guess-and-check method or your calculator to determine a possible factor for the cubic: 15x x 2 + 5x 6 P( 2) = 15( 2) ( 2) 2 + 5( 2) 6 = 0, so ( x + 2) is a factor and 2 is a value can be used in synthetic division to further break down the cubic into a quadratic Ê So, P( x) = ( x 1) ( x + 2) 15x 2 ˆ + 4x 3 Á = ( x 1) ( x + 2) ( 3x 1) ( 5x + 3) 3 P( x) = 10x x x 6 k = and 3 are polynomial functions 2, 4 and 5 are not polynomial functions 5 A 6 The minimum degree is 4, the factored form of the equation is f( x) = 1 15 ( x + 3) ( x 1) ( x 5) 2 1

8 7 C 8 C 9 V( x) = hwl = x( 10 2x) ( 20 2x) Restriction 0 < x < 5: Reason, all THREE dimensions of the box need to be positive values and the total volume of this box, represented by the y-value of the graph also has to be positive Using your calculator, determine the maximum value on the graph between 0 < x < 5; In this case x 211 cm Using your maximum from the previous question, y 192 cm C 12 D:x 3 or x 1 and R:y 0 13 When y = f( x) is transformed to y = f( x), the invariant points exist where f( x) = 0 and f( x) = 1 Thus when you put these values into the equation for f( x), the invariant points are located at Ê Á 6,0 ˆ and Ê Á 8,1 ˆ 14 x-intercept is 175, which is the same as the zero of the function 2

9 15 16 The horizontal asymptote is at y = 15 and R: y ò, y There is a point of discontinuity when x = 7 (as you cannot divide by 0), so substitute that value of x into ( 2x 1) since f( x) = ( 2x 1); x 7 thus f( 7) = 13, and the point of discontinuity is at Ê Á7,13ˆ C 20 B 3

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

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