Practice Polynomial Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A polynomial function is formed by adding terms of the form $ax^n$ where $n$ is a non-negative integer. The highest power determines the degree, which controls the graph's end behavior, maximum turning points, and number of possible real zeros.

Sums of power terms with whole-number exponents. The building blocks of functions.

Showing a random 20 of 50 problems.

Example 1

challenge

f(x)=x^3+ax+b

has

x=1

as a double root, find

a

and

b

Example 2

easy

h(x) = \frac{3}{x} + x^2

a polynomial? Why or why not?

Example 3

medium

Find all rational roots of

p(x) = 2x^3 - 3x^2 - 3x + 2

Example 4

medium

Find the

y

-intercept and one

x

-intercept of

f(x)=(x-1)(x+4)

Example 5

hard

Show that

f(x) = x^4 + x^2 + 1

has no real zeros.

Example 6

medium

Add

(3x^3 - 2x + 1) + (x^3 + 4x^2 - x + 5)

and state the degree of the result.

Example 7

medium

Find the real roots of

f(x)=x^2-5x+6

Example 8

easy

What is the leading coefficient of

p(x) = 6 - 8x^2 + 3x^4

Example 9

easy

What is the end behavior of

f(x)=x^2

x\to\pm\infty

Example 10

medium

Find the sum of the coefficients of

f(x)=2x^3-x^2+4x-3

Example 11

easy

How many roots can a degree-

3

polynomial have at most?

Example 12

medium

Factor

f(x) = x^3 - x

completely and list all real zeros.

Example 13

easy

Find the constant term of

f(x)=x^3+4x-9

Example 14

medium

Use Descartes' Rule of Signs to determine the maximum number of positive real zeros of

f(x) = x^4 - 3x^3 + 2x - 5

Example 15

easy

State the degree of

g(x) = 7 - 4x + x^5 - 2x^3

Example 16

easy

Describe the end behavior of

f(x) = x^4 - 5x^2 + 1

x \to \pm\infty

Example 17

medium

Perform synthetic division of

f(x) = 2x^3 + x^2 - 8x + 3

x - 2

Example 18

hard

f(x) = x^3 + ax^2 + bx + c

has zeros

1, 2, 3

, find

a, b, c

Example 19

medium

Find the maximum number of turning points of a degree-

5

polynomial.

Example 20

medium

Find all zeros of

p(x) = x^3 - 4x^2 + x + 6

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Related Concepts

Variables Exponents Quadratic Functions Factoring