$f(x) \to -\infty$ as $x \to \pm \infty$

Question 1

Which of the following is a possible rational zero of the polynomial $f(x) = x^3 - x^2 - 37x + 35$ according to the Rational Root Theorem?

Accepted Answer

$\pm 1, \pm 35, \pm 5, \pm 7$

Question 2

Given $f(x) = x^2 - 4x + 3$, which of the following is a zero of $f(x)$?

Accepted Answer

$x = 1$

Question 3

If $f(x) = x^3 - 3x^2 + 3x - 1$, what is the end behavior of the graph as $x 	o \infty$?

Accepted Answer

$f(x) 	o \infty$

Question 4

Which of the following polynomials has a double root at $x = -4$?

Accepted Answer

$(x+4)^2$

Question 5

Given $f(x) = x^3 + 2x^2 - 5x - 6$, use synthetic division to determine if $x = -3$ is a zero of $f(x)$. What is the remainder?

Accepted Answer

$0$

Question 6

For the polynomial $f(x) = x^4 - 8x^2 + 16$, which of the following describes the number and type of real zeros?

Accepted Answer

Two real zeros, each with multiplicity 2

Question 7

If $f(x) = x^3 - 6x^2 + 11x - 6$, which of the following is NOT a zero of $f(x)$?

Accepted Answer

$x = 4$

Question 8

Given $f(x) = x^2 + 4x + 5$, what is the vertex of the parabola?

Accepted Answer

$( -2, 1 )$

Question 9

Which of the following describes the end behavior of $f(x) = -2x^4 + 5x^2 - 1$?

Accepted Answer

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

Question 10

Given the rational function $f(x) = \frac{x^2 - 9}{x^2 - x - 6}$, which of the following is a vertical asymptote?

Accepted Answer

$x = 2$

Question 11

For $f(x) = x^3 - 2x^2 - x + 2$, which of the following is a factor of $f(x)$?

Accepted Answer

$x + 1$

Question 12

If $f(x) = x^2 - 6x + 8$, what is the sum of the zeros of $f(x)$?

Accepted Answer

$6$

Question 13

Given $f(x) = x^2 - 2x + 5$, what is the discriminant and what does it tell you about the zeros?

Accepted Answer

Discriminant is $-16$, so there are two complex zeros

Question 14

Which of the following is the correct vertex form for $f(x) = x^2 - 6x + 18$?

Accepted Answer

$f(x) = (x-3)^2 + 9$

Question 15

For the rational function $f(x) = \frac{(x+1)(3x-2)}{(x-3)(x+3)}$, what is the domain?

Accepted Answer

$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$

Question 16

If $f(x) = x^3 - 4x^2 + 5x - 2$, use the Intermediate Value Theorem to determine if there is a zero between $x = 0$ and $x = 1$.

Accepted Answer

Yes, because $f(0)$ and $f(1)$ have opposite signs

Question 17

Given $f(x) = x^4 - 5x^2 + 4$, which of the following is a real zero?

Accepted Answer

$x = 1$

Question 18

For the polynomial $f(x) = x^3 - 2x^2 - x + 2$, what is the sum of the coefficients?

Accepted Answer

$0$

Question 19

Which of the following describes the slant asymptote of $f(x) = \frac{x^2 + 5x + 10}{x + 3}$?

Accepted Answer

$y = x + 2$

Question 20

If $f(x) = x^3 - 6x^2 + 11x - 6$, which of the following is the factorization of $f(x)$?

Accepted Answer

$(x-1)(x-2)(x-3)$

Polynomial and Rational Functions: Zeros, Factorization, and Graph Behavior

Study Guide - Practice Questions