Multiplying Polynomials

Learn how to multiply polynomials using the distributive property and FOIL method.

Intermediate25 minLesson
Printable worksheet

Definition

Multiplying polynomials means distributing each term of one polynomial to every term of the other polynomial, then combining like terms.
The distributive property states:
a(b+c)=ab+aca(b + c) = ab + ac
For two binomials, we use the FOIL method:
(a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd
FOIL stands for:
  • First terms: a×ca \times c
  • Outer terms: a×da \times d
  • Inner terms: b×cb \times c
  • Last terms: b×db \times d

Try it now

What is 2x(x+3)2x(x + 3)?

Worked Examples

Multiply: 3x(2x+5)3x(2x + 5)
1

Distribute the monomial to each term

3x×2x+3x×53x \times 2x + 3x \times 5 → Apply distributive property
2

Multiply the first product

3x×2x=6x23x \times 2x = 6x^2
3

Multiply the second product

3x×5=15x3x \times 5 = 15x
4

Combine the terms

6x2+15x6x^2 + 15x → Final answer

Common Mistakes

Only multiplying the first and last terms: (x+3)(x+4)=x2+12(x + 3)(x + 4) = x^2 + 12

Why it's wrong: Every term in the first polynomial must multiply every term in the second polynomial.

Correct: Use FOIL: x2+4x+3x+12=x2+7x+12x^2 + 4x + 3x + 12 = x^2 + 7x + 12

Forgetting the middle term when squaring: (x+5)2=x2+25(x + 5)^2 = x^2 + 25

Why it's wrong: Squaring a binomial creates three terms, not two. The middle term is 2ab2ab.

Correct: (x+5)2=x2+10x+25(x + 5)^2 = x^2 + 10x + 25

Sign errors with negative terms: (−3)(5)=15(-3)(5) = 15

Why it's wrong: A negative times a positive is negative.

Correct: (−3)(5)=−15(-3)(5) = -15

Adding exponents incorrectly: x×x=2xx \times x = 2x

Why it's wrong: When multiplying variables, we add exponents: x1×x1=x2x^1 \times x^1 = x^2

Correct: x×x=x2x \times x = x^2

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Multiply Polynomials | Step-by-Step Examples

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Interactive Visual

Area model: multiplying binomials

The rectangle is (x + a) by (x + b). Its four pieces are the four products of FOIL.

x2x^2
3x3x
4x4x
1212
xx
33
xx
44

Length × width = area

(x+3)(x+4)=x2+7x+12(x + 3)(x + 4) = x^2 + 7x + 12

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is 2x(x+3)2x(x + 3)?

Why It Matters

Multiplying polynomials is essential for:
  • Factoring: To factor, you need to understand how factors multiply
  • Solving equations: Quadratic equations often come from polynomial multiplication
  • Area calculations: Finding the area of rectangles with variable dimensions
  • Real-world modeling: Physics, engineering, and economics use polynomial products
Mastering this skill opens doors to more advanced algebra topics!

Real World Applications

Area of a Garden

Finding areas with variable dimensions requires polynomial multiplication.

Example:

A rectangular garden has length (x+5)(x + 5) meters and width (x+3)(x + 3) meters. The area is (x+5)(x+3)=x2+8x+15(x + 5)(x + 3) = x^2 + 8x + 15 square meters.

1Try It Yourself

A frame around a picture adds 2 cm to each side. The picture is xx cm by xx cm.

What is the total area including the frame?

Step 1: Write the mathematical expression

Multiply (x+4)(x+4)(x + 4)(x + 4):

Business Revenue

Revenue calculations often involve multiplying expressions for price and quantity.

Example:

If price is (20−x)(20 - x) dollars and quantity sold is (100+5x)(100 + 5x) items, revenue is (20−x)(100+5x)=2000+100x−100x−5x2=2000−5x2(20 - x)(100 + 5x) = 2000 + 100x - 100x - 5x^2 = 2000 - 5x^2 dollars.

2Try It Yourself

A shop sells (50+x)(50 + x) items at (10−x)(10 - x) euros each.

Write the revenue as a polynomial.

Step 1: Write the mathematical expression

Multiply (50+x)(10−x)(50 + x)(10 - x):

Key Takeaways

  • 1Use the distributive property to multiply each term in one polynomial by every term in the other
  • 2For two binomials, use FOIL: First, Outer, Inner, Last
  • 3Always combine like terms after multiplying
  • 4When squaring a binomial: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • 5Watch for sign errors especially with negative terms

Frequently Asked Questions

FOIL only works for multiplying two binomials. For other cases, use the general distributive property.
FOIL only works for multiplying two binomials. For other cases, use the general distributive property.
Squaring means multiplying by itself: (x+3)(x+3)(x + 3)(x + 3). Using FOIL gives x2+3x+3x+9=x2+6x+9x^2 + 3x + 3x + 9 = x^2 + 6x + 9.
Combine terms with the same variable and exponent. Stop when no more like terms remain.

Glossary

Polynomial
An expression with one or more terms involving variables and coefficients (e.g., 3x2+2x−53x^2 + 2x - 5)
Binomial
A polynomial with exactly two terms (e.g., x+3x + 3)
Trinomial
A polynomial with exactly three terms (e.g., x2+5x+6x^2 + 5x + 6)
FOIL
A method for multiplying two binomials: First, Outer, Inner, Last
Like terms
Terms with the same variable raised to the same power (e.g., 3x23x^2 and 5x25x^2)

Formula Card

Distributive Property

a(b+c)=ab+aca(b + c) = ab + ac

Multiply the term outside by each term inside

FOIL Method

(a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

First, Outer, Inner, Last for two binomials

Square of a Sum

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Perfect square trinomial pattern

Square of a Difference

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Note the minus sign on middle term

Difference of Squares

(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

Middle terms cancel out

Today's challengeOne problem, three tries

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