Polynomial Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polynomial Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A polynomial function adds terms $ax^n$ with non-negative integer exponents, and its degree governs its shape.

Common stuck point: The procedure for polynomial functions is the easy part; the trap is allowing negative or fractional exponents. Asking "Is every term a constant times $x$ to a whole-number power, with only addition and subtraction joining them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every term a constant times $x$ to a whole-number power, with only addition and subtraction joining them?

Worked Examples

Example 1

easy

Find the degree and leading coefficient of

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

Answer

\text{Degree } 4, \quad \text{leading coefficient } -3

First step

Write the polynomial in descending powers and identify the highest exponent present.

Full solution

2
The highest power is $x^4$ , so the degree is $4$ .
3
The coefficient attached to the leading term $-3x^4$ is $-3$ , so the leading coefficient is $-3$ .

The degree determines the end behavior and maximum number of roots. A negative leading coefficient with even degree means the graph falls on both ends.

Example 2

medium

Find all zeros of

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

Example 3

medium

Factor

f(x) = x^3 - x

completely and list all real zeros.

Example 4

medium

Perform synthetic division of

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

x - 2

Example 5

hard

Show that

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

has no real zeros.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Perform polynomial long division:

\frac{2x^3 + 3x^2 - 5x + 1}{x + 2}

Example 2

hard

Sketch the end behavior and find all real zeros of

f(x) = -2x^3 + 6x^2 + 8x

. State the multiplicity of each zero.

Example 3

easy

What is the degree of

f(x)=3x^4-2x+7

Example 4

easy

What is the leading coefficient of

f(x)=-5x^3+2x^2-1

Example 5

easy

Evaluate

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

x=2

Example 6

easy

f(x)=2x^{1/2}+1

a polynomial?

Example 7

easy

Find the constant term of

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

Example 8

easy

How many roots can a degree-

3

polynomial have at most?

Example 9

easy

What is the end behavior of

f(x)=x^2

x\to\pm\infty

Example 10

easy

Factor

f(x)=x^2-9

Example 11

medium

Find the real roots of

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

Example 12

medium

Describe the end behavior of

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

Example 13

medium

How many real roots does

f(x)=x^2+4

have?

Example 14

medium

Find the maximum number of turning points of a degree-

5

polynomial.

Example 15

medium

x=3

is a root of

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

, factor it fully.

Example 16

medium

Evaluate

f(2)

for

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

using direct substitution.

Example 17

medium

Find the

y

-intercept and one

x

-intercept of

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

Example 18

medium

A polynomial has roots

2

and

-3

and leading coefficient

1

. Write it.

Example 19

challenge

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

has

x=1

as a double root, find

a

and

b

Example 20

challenge

The polynomial

x^3-6x^2+11x-6

has roots summing to what value?

Example 21

challenge

Find the remainder when

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

is divided by

x-2

Example 22

medium

Find the sum of the coefficients of

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

Example 23

easy

State the degree of

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

Example 24

easy

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

a polynomial? Why or why not?

Example 25

easy

Evaluate

p(-1)

for

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

Example 26

easy

Describe the end behavior of

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

x \to \pm\infty

Example 27

medium

Use the Remainder Theorem to find the remainder when

f(x) = x^3 - 2x^2 + 4x - 7

is divided by

x - 3

Example 28

medium

Find all rational roots of

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

Example 29

medium

Given

f(x) = x^3 + kx + 2

has

x = -1

as a root, find

k

Example 30

medium

Write a polynomial of least degree with real coefficients, leading coefficient

1

, and zeros

0

2

, and

-3

Example 31

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 32

medium

State the multiplicity of the zero

x = 2

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

Example 33

medium

Add

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

and state the degree of the result.

Example 34

hard

Find all real zeros of

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

Example 35

hard

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

has zeros

1, 2, 3

, find

a, b, c

Example 36

hard

Find the equation of a degree-

3

polynomial with real coefficients, leading coefficient

1

, that has zeros

1

and

2+i

Example 37

hard

Find the value of

f(10)

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

using synthetic substitution.

Example 38

hard

For what values of

k

does

f(x) = x^3 - 3x + k

have exactly one real root?

Example 39

challenge

p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1

, write

p(x)

in a simpler closed form.

Example 40

challenge

A polynomial

f(x)

of degree

3

satisfies

f(0)=1, f(1)=2, f(2)=5, f(3)=10

. Find

f(4)

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Related Concepts

Variables Exponents Quadratic Functions Factoring

Background Knowledge

These ideas may be useful before you work through the harder examples.

variablesexponents