Question 1

Given the polynomial $$f(x) = 6x^5$ - 12x^4$ + 20x^3$ - 7x + 1$$, what is the degree of the polynomial?

Accepted Answer

$$5$$

Question 2

For the polynomial $$f(x) = 6x^5$ - 12x^4$ + 20x^3$ - 7x + 1$$, what is the leading term?

Accepted Answer

$$6x^5$$

Question 3

What is the leading coefficient of the polynomial $$f(x) = 6x^5$ - 12x^4$ + 20x^3$ - 7x + 1$$?

Accepted Answer

$$6$$

Question 4

Find the zeros and their multiplicities for $$g(x) = (x+1)^3$(x-4)(x+7)^2$.

Accepted Answer

Zero at $$x = -1 ($$multiplicity 3), $$x = 4 ($$multiplicity 1), $$x = -7 ($$multiplicity 2)

Question 5

Using synthetic division, what is the remainder when $$x^3$$ - $$7x^2 + 13x - 3$ is divided by x - 2$?

Accepted Answer

$$1$$

Question 6

List all possible rational zeros for $$h(x) = 3x^4$ - 5x^3$ + x^2$ - x + 1$$ using the Rational Root Theorem.

Accepted Answer

Possible zeros: $$\pm1, \pm3, \pm1/3$$

Question 7

Find a polynomial function of degree 5 with $$-1 as a $$zero of multiplicity 3 and $$0 as a $$zero of multiplicity 2.

Accepted Answer

$$f(x) = (x+1)^3$x^2$

Question 8

Find all vertical and horizontal asymptotes for $$f(x) = \frac{4x^2$ + 5x}{x^2$ + 32x - 4}.$$

Accepted Answer

Vertical: $$x^2 + 32x - 4 = 0$; Horizontal: y = 4$

Question 9

Find the $$x- $$and $$y-$$intercepts for $$f(x) = \frac{3x^2$ - 2x}{x^2$ - x + 2}.$$

Accepted Answer

$$x-$$intercepts: $$x = 0, x = \frac{2}{3}$$; $$y-$$intercept: $$y = 0$$

Question 10

Solve the inequality $$x^2 + 2x$$ \u003c 0$$ and write the solution in interval notation.

Accepted Answer

$$(-2, 0)$$

Question 11

Which of the following is the correct solution to $$x^2$$ \geq 25$$ in interval notation?

Accepted Answer

$$(-\infty, -5] \cup [5, \infty)$$

Question 12

Given $$h(x) = 2(x-4)^3$(x-4)^2(x-6$$)$, what is the degree of the polynomial?

Accepted Answer

$$6$$

Question 13

Using synthetic division, determine if $$x = -3 is a $$zero of $$f(x) = 3x^2$ - 5x + 6x - 18.$$

Accepted Answer

No, because the remainder is not $$0$$

Question 14

Find a polynomial function of degree 3 with zeros at $$2$$, $$-1$$, and $$0$$, each with multiplicity 1.

Accepted Answer

$$f(x) = x(x-2)(x+1)$$

Question 15

Solve the inequality $$x^2 - 4x + 3$$ \u003c 0$$ and state the critical values.

Accepted Answer

Critical values: $$x = 1, x = 3$$; Solution: $$(1, 3)$$

Question 16

Which of the following is the correct solution to $$x^3 + 7x$$ \u003c 0$$?

Accepted Answer

$$(-\infty, 0)$$

Question 17

Using synthetic division, what is the quotient when $$x^4 - 36$ is divided by x + 3$?

Accepted Answer

$$x^3$$ - $$3x^2 + 9x - 27$

Question 18

List all possible rational zeros for $$h(x) = 6x^3$ - 2x^2$ + 7x - 18.$$

Accepted Answer

Possible zeros: $$\pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm1/6, \pm3/6, \pm9/6$$

Question 19

Solve the inequality $$x^2 - 2x - 3$$ \geq 5$$ and write the solution in interval notation.

Accepted Answer

$$(-\infty, -2] \cup [4, \infty)$$

College Algebra Unit 3 Polynomial and Rational Functions Review – Step-by-Step Guidance

Study Guide - Practice Questions