Q: Which of the following describes the end behavior of $$f(x) = x^5$ - 2x^3$ + x$$?

As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Question 1

What is the degree and leading coefficient of the polynomial $$f(x) = -2x^4$ + 3x^2$ - 7x + 5$$?

Accepted Answer

Degree: 4, Leading coefficient: $$-2$$

Question 2

Which of the following is NOT a polynomial function?

Accepted Answer

$$h(x) = \dfrac{2}{x}$$

Question 3

Given $$f(x) = (x+2)(x-5)(x-1)$$, what are the zeros and their multiplicities?

Accepted Answer

Zeros: $$-2, 5, 1$$; all multiplicity 1

Question 4

If a polynomial $$f(x)$$ has a zero at $$x = 4$$ with multiplicity 2, what does the graph do at $$x = 4$$?

Accepted Answer

It touches and bounces off the x-axis at $$x = 4$$

Question 5

Use the Intermediate Value Theorem to show that $$f(x) = x^3$ - 5x^2$ + 2x + 8$$ has a zero between $$x = 2$$ and $$x = 3.$$

Accepted Answer

$$f(2) = -4$$, $$f(3) = 2$$, so there is a zero between them.

Question 6

What is the end behavior of $$f(x) = -3x^4$ + 2x^2$ - 1 as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o -\infty$$

Question 7

Given the polynomial $$f(x) = x^3$ - 4x^2$ + x + 6$$, use synthetic division to divide by $$x - 2. $$What is the remainder?

Accepted Answer

Remainder is $$-2$$

Question 8

Which of the following is a possible rational zero of $$f(x) = 3x^3$ + 5x^2$ - 8x + 10$$ according to the Rational Zero Theorem?

Accepted Answer

$$-2$$

Question 9

If $$f(x) = x^4$ - 2x^2$ + x - 3$$, what is the degree and how many possible real zeros does it have?

Accepted Answer

Degree: 4, up to 4 real zeros

Question 10

Divide $$4x^3$$ + $$3x^2 - 8x + 2$ by x + 3$ using synthetic division. What is the quotient?

Accepted Answer

$$4x^2 - 9x + 19$

Question 11

Given the graph of a polynomial with zeros at $$x = -2, 0, 3$$ and the zero at $$x = 3$$ has multiplicity 2, which equation could represent the polynomial?

Accepted Answer

$$f(x) = (x + 2)x(x - 3)^2$

Question 12

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, solve for all real zeros.

Accepted Answer

$$x = 1, 2, 3$$

Question 13

What is the remainder when $$f(x) = 3x^4$ + 4x^3$ - 5x + 3 is $$divided by $$x - 2$$?

Accepted Answer

$$105$$

Question 14

Which of the following describes the end behavior of $$f(x) = x^5$ - 2x^3$ + x$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 15

Given $$f(x) = x^3$ + 4x^2$ + x - 6$$, use synthetic division to find all zeros.

Accepted Answer

$$x = 1, -2, -3$$

Question 16

If $$f(x) = x^4$ + 2x^3$ - 7x^2$ + 8x - 2$$, which of the following is a possible rational zero?

Accepted Answer

$$1$$

Question 17

A polynomial $$f(x)$$ has zeros at $$x = -1, 2, 5$$ with $$x = 2$$ having multiplicity 2. Which statement is true about the graph at $$x = 2$$?

Accepted Answer

The graph touches and bounces off the x-axis at $$x = 2$$

Question 18

Given $$f(x) = x^3$ - 2x^2$ + 4x - 8$$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2, \pm4, \pm8$$

Precalculus Polynomial Functions and Theorems Study Guide

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