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

Zero: $x = 6$, Multiplicity: $5$

$f(2) 0$, so there is a zero between $x = 2$ and $x = 3$

Question 1

What is the vertex of the parabola given by $f(x) = -7x^2 + 3$?

Accepted Answer

$(0, 3)$

Question 2

Find the axis of symmetry for the parabola $f(x) = 3x^2 + 3$.

Accepted Answer

$x = 0$

Question 3

Determine the domain and range of the function $f(x) = x^2 - 2$.

Accepted Answer

Domain: $(-\infty, \infty)$; Range: $[-2, \infty)$

Question 4

Which of the following polynomial functions has a graph symmetric with respect to the y-axis?

Accepted Answer

$y = 4x^2 + 2$

Question 5

Determine if the function $f(x) = x^4 - 2x^2 - 3$ is even, odd, or neither.

Accepted Answer

Even

Question 6

What is the leading term, leading coefficient, and degree of the polynomial $f(x) = 7x^8 - 10 + 0.12x - x^3$?

Accepted Answer

Leading term: $7x^8$, Leading coefficient: $7$, Degree: $8$

Question 7

Use synthetic division to divide $3x^2 - 3x + 6$ by $x + 2$. What is the remainder?

Accepted Answer

$6$

Question 8

Given $f(x) = 2x^3 + 5x^2 + 12x - 8$, which of the following is NOT a possible rational zero according to the Rational Root Theorem?

Accepted Answer

$\frac{3}{2}$

Question 9

Use Descartes' Rule of Signs to determine the possible number of positive real zeros for $f(x) = 8x^3 - 6x^2 + 8x + 8$.

Accepted Answer

2 or 0

Question 10

Find all rational zeros and factor $f(x) = x^3 + 2x^2 - 16x - 32$.

Accepted Answer

Rational zeros: $-2, -4, 4$; Factored form: $(x + 2)(x + 4)(x - 4)$

Question 11

State the domain of the rational function $f(x) = \frac{7x - 6}{8x - 16}$.

Accepted Answer

$x 
eq 2$

Question 12

Find the vertical asymptote(s) of $f(x) = \frac{x + 9}{x^2 - 9}$.

Accepted Answer

$x = 3$ and $x = -3$

Question 13

Find the horizontal asymptote of $f(x) = \frac{3x^2 - 4x - 6}{5x^2 - 5x + 5}$.

Accepted Answer

$y = \frac{3}{5}$

Question 14

Find the slant (oblique) asymptote of $f(x) = \frac{x^2 + 4x + 3}{x - 9}$.

Accepted Answer

$y = x - 5$

Question 15

Solve the inequality $\frac{x - 3}{x^2 - 1} < 0$ and describe the solution set.

Accepted Answer

$(-\infty, -1) \cup (1, 3)$

Question 16

Which of the following describes the end behavior of $f(x) = 7x^3 - 3x^2 - 5x + 1$?

Accepted Answer

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

Question 17

Given $f(x) = 5(x - 6)^2(x - 6)^3$, what is the zero and its multiplicity?

Accepted Answer

Zero: $x = 6$, Multiplicity: $5$

Question 18

Use the intermediate value theorem to show that $f(x) = x^3 + 2x^2 - 16x - 32$ has a real zero between $x = 2$ and $x = 3$.

Accepted Answer

$f(2) < 0$ and $f(3) > 0$, so there is a zero between $x = 2$ and $x = 3$

Question 19

Solve the inequality $5x < 8x$ and graph the solution set.

Accepted Answer

$x > 0$

Polynomial and Rational Functions: Key Concepts and Problem-Solving Strategies

Study Guide - Practice Questions