A polynomial of degree $n$ has exactly $n$ complex zeros, counting multiplicities.

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

$f(x) = \dfrac{2x^2 + 1}{x^3 - 4}$

Question 1

Which of the following expressions is a polynomial?

Accepted Answer

$f(x) = 4x^2 - 7x + 9$

Question 2

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

Accepted Answer

Degree: 4, Leading coefficient: 2

Question 3

Which of the following is the correct standard form of the equation of a circle with center $(3, -2)$ and radius $5$?

Accepted Answer

$(x - 3)^2 + (y + 2)^2 = 25$

Question 4

Solve the inequality $2x - 5 < 7$.

Accepted Answer

$x < 4$

Question 5

If $f(x) = x^3 - 4x^2 + x + 6$ is divided by $x - 2$, what is the remainder?

Accepted Answer

$2$

Question 6

Which of the following is a rational function?

Accepted Answer

$f(x) = \dfrac{3x + 2}{x - 1}$

Question 7

A circle has the equation $x^2 + y^2 - 6x + 8y = 0$. What are the center and radius of the circle?

Accepted Answer

Center: $(3, -4)$, Radius: $5$

Question 8

Which of the following statements about the Fundamental Theorem of Algebra is correct?

Accepted Answer

A polynomial of degree $n$ has exactly $n$ complex zeros, counting multiplicities.

Question 9

Solve the radical equation: $\sqrt{x + 5} = x - 1$.

Accepted Answer

$x = 3$

Question 10

Which of the following describes inverse variation?

Accepted Answer

$y = \dfrac{k}{x}$

Question 11

Given the rational function $f(x) = \dfrac{2x^2 - 3x + 1}{x^2 - 4}$, what are the vertical asymptotes?

Accepted Answer

$x = 2$ and $x = -2$

Question 12

If $y$ varies directly as $x$ and $y = 10$ when $x = 2$, what is the value of $y$ when $x = 5$?

Accepted Answer

$y = 15$

Question 13

Which of the following is NOT a possible rational zero of $f(x) = 6x^3 - 5x^2 + 2x - 1$ according to the Rational Zero Theorem?

Accepted Answer

$\dfrac{2}{3}$

Question 14

Solve the rational equation: $\dfrac{x}{x-2} = 3$.

Accepted Answer

$x = 6$

Question 15

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

Accepted Answer

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

Question 16

Which of the following is the correct equation of a circle with diameter endpoints at $(1, 2)$ and $(5, 6)$?

Accepted Answer

$(x - 3)^2 + (y - 4)^2 = 8$

Question 17

If $f(x) = x^2 - 4x + 3$, what are the $x$-intercepts of the graph?

Accepted Answer

$x = 1$ and $x = 3$

Question 18

Which of the following is the solution set for the inequality $x^2 - 9 < 0$?

Accepted Answer

$-3 < x < 3$

Question 19

A rational function has a horizontal asymptote at $y = 0$. Which of the following could be its equation?

Accepted Answer

$f(x) = \dfrac{2x^2 + 1}{x^3 - 4}$

College Algebra: Study Guide for Polynomial, Rational, Radical Functions, and Circles

Study Guide - Practice Questions