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 axnax^n where nn 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
If f(x)=x3+ax+bf(x)=x^3+ax+b has x=1x=1 as a double root, find aa and bb.

Example 2

easy
Is h(x)=3x+x2h(x) = \frac{3}{x} + x^2 a polynomial? Why or why not?

Example 3

medium
Find all rational roots of p(x)=2x3−3x2−3x+2p(x) = 2x^3 - 3x^2 - 3x + 2.

Example 4

medium
Find the yy-intercept and one xx-intercept of f(x)=(x−1)(x+4)f(x)=(x-1)(x+4).

Example 5

hard
Show that f(x)=x4+x2+1f(x) = x^4 + x^2 + 1 has no real zeros.

Example 6

medium
Add (3x3−2x+1)+(x3+4x2−x+5)(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)=x2−5x+6f(x)=x^2-5x+6.

Example 8

easy
What is the leading coefficient of p(x)=6−8x2+3x4p(x) = 6 - 8x^2 + 3x^4?

Example 9

easy
What is the end behavior of f(x)=x2f(x)=x^2 as x→±∞x\to\pm\infty?

Example 10

medium
Find the sum of the coefficients of f(x)=2x3−x2+4x−3f(x)=2x^3-x^2+4x-3.

Example 11

easy
How many roots can a degree-33 polynomial have at most?

Example 12

medium
Factor f(x)=x3−xf(x) = x^3 - x completely and list all real zeros.

Example 13

easy
Find the constant term of f(x)=x3+4x−9f(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)=x4−3x3+2x−5f(x) = x^4 - 3x^3 + 2x - 5.

Example 15

easy
State the degree of g(x)=7−4x+x5−2x3g(x) = 7 - 4x + x^5 - 2x^3.

Example 16

easy
Describe the end behavior of f(x)=x4−5x2+1f(x) = x^4 - 5x^2 + 1 as x→±∞x \to \pm\infty.

Example 17

medium
Perform synthetic division of f(x)=2x3+x2−8x+3f(x) = 2x^3 + x^2 - 8x + 3 by x−2x - 2.

Example 18

hard
If f(x)=x3+ax2+bx+cf(x) = x^3 + ax^2 + bx + c has zeros 1,2,31, 2, 3, find a,b,ca, b, c.

Example 19

medium
Find the maximum number of turning points of a degree-55 polynomial.

Example 20

medium
Find all zeros of p(x)=x3−4x2+x+6p(x) = x^3 - 4x^2 + x + 6.
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