Chapter 07 of 10

Polynomials

How do algebraic expressions behave when they contain many powers?

A polynomial is a sum of terms with whole-number exponents. Its degree describes its highest power and helps predict its behavior. Polynomial operations extend the same distribution and like-term ideas used earlier.

01

The big ideas

Understand these before you worry about speed.

Classify by degree and terms

The degree is the greatest exponent after simplification. Monomial, binomial, and trinomial describe the number of terms.

Example4x³ − x + 2 is a cubic trinomial.

Add and subtract like terms

Align matching powers. When subtracting a polynomial, distribute the negative sign to every term.

Example(3x² + x) − (x² − 4x) = 2x² + 5x.

Multiply by distributing

Every term in one factor multiplies every term in the other. Then combine like terms.

Example(x + 2)(x + 5) = x² + 7x + 10.
02

Worked example

Follow the reason for each step—not just the symbols.

Problem

Multiply (2x − 3)(x + 4).

  1. Multiply 2x by both terms: 2x² + 8x.
  2. Multiply −3 by both terms: −3x − 12.
  3. Combine like terms: 8x − 3x = 5x.
Answer2x² + 5x − 12
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Find the degree of 7x⁴ − 2x² + 1.

Check answer4
2

Add (2x² + 3x − 1) + (x² − 5x + 4).

Check answer3x² − 2x + 3
3

Multiply 3x(2x² − x + 5).

Check answer6x³ − 3x² + 15x
4

Multiply (x − 6)(x + 2).

Check answerx² − 4x − 12
04

Practice and tools

Use the resource that matches what you need next.

Worksheet

Polynomial Operations

Add, subtract, distribute, and multiply polynomials.

Open resource Worksheet

Polynomial Graphs

Connect degree and leading coefficient to graph shape.

Open resource Quiz

Algebra 1 Final Test

Check polynomial skills as part of the complete 20-question course assessment.

Open resource