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Exam 3: Polynomial and Rational Functions

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Question 21

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Use the rational roots theorem to write the function in factored form and find all zeroes. Note a = 1. p(x) = x⁴ + 6x³ - 13x² - 66x + 72

A) p(x) = (x - 1) (x - 3) (x + 4) (x + 6) ; zeroes: 1, 3, -4, -6
B) p(x) = (x + 1) (x + 3) (x - 4) (x - 6) ; zeroes: -1, -3, 4, 6
C) p(x) = (x - 1) (x + 3) (x - 4) (x + 6) ; zeroes: 1, -3, 4, -6
D) p(x) = (x + 1) (x - 3) (x + 4) (x - 6) ; zeroes: -1, 3, -4, 6

E) A) and C)
F) All of the above

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Question 22

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Gather information on the polynomial using the rational roots theorem, testing for 1 and -1, Descartes's rule of signs, and the upper and lower bounds property. Respond explicitly to each. p(x) = x⁴ - 3x³ + 9x - 27

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Possible rational roots: {±1, ...

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Question 23

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Give the location of the vertical asymptote(s) if they exist, and state whether function values will change sign (positive to negative or negative to positive) from one side of the asymptote to the other.

Find the vertex. f(x) = x² + 10x - 8

State the end behavior and y-intercept of the function. Do not graph. f(x) = x⁵ + 5x⁴ - 5x² - 3x + 5

Determine the location of the horizontal asymptote if it exists. Then determine whether the graph will cross the asymptote, and if so, where it crosses.

Find the intercepts. Round to the nearest tenth if necessary. f(x) = 3x² + 8x - 6

Solve the inequality using a number line and the behavior of the graph at each zero. Write your answer in interval notation.

Use the following to answer questions : y varies inversely with x; y = 12,600 when x = 0.2. -The days required to complete a project varies inversely with the number of employees working on the project. It takes 88 days to complete the project if 5 employees work on it. (a) What is the constant of variation? (b) Write the variation equation. (c) How long will it take to complete the project if 22 employees work on it?

Sketch the graph of the function using the degree, end behavior, x- and y-intercepts, zeroes of multiplicity, and a few midinterval points to round out the graph. Connect all points with a smooth, continuous curve. f(x) = (x - 1)³(x + 2)(x + 3)

Determine the location of the horizontal asymptote if it exists. Then determine whether the graph will cross the asymptote, and if so, where it crosses.

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Use the graph to estimate the zeroes of the function, then state whether their multiplicity is even or odd.

Find the intercepts. f(x) = x² - 10x + 24

Use the following to answer questions : (Gridlines are spaced one unit apart.) -Give the equation related to the graph. Assume |a| = 1.

Use synthetic division and the remainder theorem to show that x = -2 is a zero of P(x) = x³ - 4x² - 4x + 16.

Use the vertex/intercept formula, , to find zeroes, real or complex of the formula. f(x) = 3(x + 1) ² - 2

Use the Guidelines for Graphing Rational Functions to graph .

Graph the function using the Guidelines for Graphing Rational Functions, which is simply modified to include nonlinear asymptotes. Clearly state all intercepts and asymptotes and any additional points used to sketch the graph.

Find the zeroes of the polynomial using any combination of the rational roots theorem, synthetic division, testing for 1 and -1, and/or the remainder and factor theorems. p(x) = 4x⁴ - 3x³ - 30x² - x + 30

Use the following to answer questions : (Gridlines are spaced one unit apart.) -State the equations of the horizontal and vertical asymptotes.