Polynomials Examples in Math

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

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

Concept Recap

An expression built by adding terms that consist of constants multiplied by variables raised to non-negative integer powers.

A sum of terms like 3x2+2x−53x^2 + 2x - 5. The highest power is the degree.

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 adds terms of the form (number)×xwhole number\times x^{\text{whole number}}, like 3x2+2x−53x^2+2x-5.

Common stuck point: The procedure for polynomials is the easy part; the trap is counting an expression with a negative or fractional exponent as a polynomial. Asking "Is every exponent on the variable a whole number ≥0\ge 0 with no variable in a denominator?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every exponent on the variable a whole number ≥0\ge 0 with no variable in a denominator?

Worked Examples

Example 1

easy
What is the degree of the polynomial 4x3−2x2+x−74x^3 - 2x^2 + x - 7?

Answer

Degree 33

First step

1
Identify the exponent of each term: x3x^3 has degree 3, x2x^2 has degree 2, xx has degree 1, −7-7 has degree 0.

Full solution

  1. 2
    The degree of the polynomial is the highest exponent.
  2. 3
    The degree is 3.
The degree of a polynomial is the largest power of the variable. It determines the polynomial's end behavior and the maximum number of zeros.

Example 2

medium
Add the polynomials (3x2+2x−5)(3x^2 + 2x - 5) and (x2−4x+3)(x^2 - 4x + 3).

Example 3

medium
Multiply (2x+3)(x2−x+4)(2x + 3)(x^2 - x + 4).

Example 4

medium
Multiply (x+5)(x−3)(x + 5)(x - 3).

Example 5

medium
Expand (2x−1)2(2x - 1)^2.

Example 6

medium
Expand (x+3)(x−3)(x + 3)(x - 3).

Example 7

medium
Multiply (x−2)(x2+3x−4)(x - 2)(x^2 + 3x - 4).

Example 8

medium
Find a polynomial that represents the area of a rectangle with length x+4x + 4 and width x−2x - 2.

Example 9

hard
Find the value of kk so that x=2x = 2 is a zero of P(x)=x3−4x2+kx−6P(x) = x^3 - 4x^2 + kx - 6.

Example 10

hard
If P(x)=x2+ax+bP(x) = x^2 + ax + b has zeros at x=2x = 2 and x=−5x = -5, find aa and bb.

Example 11

hard
Divide x3+2x2−5x−6x^3 + 2x^2 - 5x - 6 by x−2x - 2 using synthetic division.

Example 12

challenge
Let P(x)=x3−6x2+11x−6P(x) = x^3 - 6x^2 + 11x - 6. Find all real zeros.

Practice Problems

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

Example 1

easy
Classify 5x4−x+25x^4 - x + 2 by degree and number of terms.

Example 2

medium
Subtract (2x3−x+4)(2x^3 - x + 4) from (5x3+3x2−2)(5x^3 + 3x^2 - 2).

Example 3

easy
What is the degree of 3x4−2x+13x^4 - 2x + 1?

Example 4

easy
Is x−2+3x^{-2} + 3 a polynomial?

Example 5

easy
What is the leading coefficient of −5x3+2x2-5x^3 + 2x^2?

Example 6

easy
Add: (2x+3)+(x−5)(2x + 3) + (x - 5).

Example 7

easy
Write 5+2x2−x5 + 2x^2 - x in standard (descending) order.

Example 8

easy
How many terms does x3+2x2−x+4x^3 + 2x^2 - x + 4 have?

Example 9

easy
What is the degree of the constant polynomial 77?

Example 10

easy
Subtract: (4x−1)−(x+2)(4x - 1) - (x + 2).

Example 11

medium
Multiply: (x+2)(x+3)(x + 2)(x + 3).

Example 12

medium
Multiply: 3x(2x2−x+4)3x(2x^2 - x + 4).

Example 13

medium
What is the degree of (x2+1)(x3−x)(x^2 + 1)(x^3 - x)?

Example 14

medium
Combine: (2x2−3x+1)+(x2+3x−4)(2x^2 - 3x + 1) + (x^2 + 3x - 4).

Example 15

medium
Expand (x−4)2(x - 4)^2.

Example 16

medium
Is x+1\sqrt{x} + 1 a polynomial? Why?

Example 17

medium
Find the remainder when x2+3x+5x^2 + 3x + 5 is evaluated at x=2x = 2 (Remainder Theorem).

Example 18

challenge
Factor completely: x3−xx^3 - x.

Example 19

challenge
Divide x2+5x+6x^2 + 5x + 6 by x+2x + 2.

Example 20

challenge
If P(x)P(x) has degree 33 and Q(x)Q(x) degree 22, what is the degree of P(x)+Q(x)P(x) + Q(x)?

Example 21

medium
What is the leading term of (2x+1)(3x−4)(2x + 1)(3x - 4) without full expansion?

Example 22

medium
Evaluate the polynomial x2−2x+1x^2 - 2x + 1 at x=1x = 1.

Example 23

easy
What is the degree of 7x5−3x2+87x^5 - 3x^2 + 8?

Example 24

easy
Combine like terms: 5x2+3x−2x2+x5x^2 + 3x - 2x^2 + x.

Example 25

easy
Add: (x2−4x)+(3x2+4x−7)(x^2 - 4x) + (3x^2 + 4x - 7).

Example 26

easy
Evaluate P(x)=x3−2x+1P(x) = x^3 - 2x + 1 at x=2x = 2.

Example 27

medium
Subtract (5x2+2x−1)−(2x2−3x+4)(5x^2 + 2x - 1) - (2x^2 - 3x + 4).

Example 28

medium
Multiply 3x2(2x3−4x+5)3x^2(2x^3 - 4x + 5).

Example 29

medium
Find the degree and leading coefficient of (2x2−1)(x3+4)(2x^2 - 1)(x^3 + 4) after expansion.

Example 30

medium
Evaluate P(x)=2x3−x2+3P(x) = 2x^3 - x^2 + 3 at x=−1x = -1.

Example 31

hard
Expand (x+2)3(x + 2)^3.

Example 32

hard
Find the constant term of (2x−3)4(2x - 3)^4 when expanded.
← Back to Polynomials Formula Explained Practice Mode

Background Knowledge

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

variablesexponents