How to Multiply Polynomials: From 1 Term to 3 Terms

Multiplying polynomials means multiplying every term in one polynomial by every term in the other polynomial, then combining like terms. The distributive property works for all polynomial multiplication problems.

What are Polynomials?

A polynomial is a mathematical expression made by adding, subtracting, or multiplying numbers and variables, with exponents that are whole numbers. It has a limited number of terms. Each term can have numbers, variables (like x, y), or both, multiplied together.

Example:

3𝑥² + 2𝑥 − 5

This polynomial has three terms: 3𝑥², 2𝑥, and -5.

Multiplying Polynomials: Rules and Step-by-Step Examples

Multiplying polynomials involves multiplying each term in one polynomial by each term in another. This is an important algebra skill introduced in Pre-Algebra and developed further in Algebra 1 as students learn to work with more complex expressions.

We’ll break it down step by step, from easy to harder problems.

1 Term × 1 Term (Monomial × Monomial)

1. Multiply the numbers;
2. Multiply the variables by adding exponents.

Example:

3𝑥 × 4𝑥²

3𝑥 × 4𝑥² = 3 × 4 × 𝑥¹⁺² = 12𝑥³

1 Term × 2 Terms (Monomial × Binomial)

Multiply the single term by each term in the binomial.

Example:

2𝑥 × (𝑥 + 5)

2𝑥 × (𝑥 + 5) = 2𝑥² + 10𝑥

2 Terms × 2 Terms (Binomial × Binomial)

Use the FOIL method: First, Outer, Inner, Last. Multiply each term in the first binomial by each term in the second.

Example:

(𝑥 + 3) × (𝑦 + 2)

(𝑥 + 3) × (𝑦 + 2) = 𝑥𝑦 + 2𝑥 + 3𝑦 + 6

2 Terms × 3 Terms (Binomial × Trinomial)

Multiply each term in the first polynomial by each term in the second one.

Example:

(𝑥 + 4) × (𝑥² + 2𝑥 + 1)

(𝑥 + 4) × (𝑥² + 2𝑥 + 1) = 𝑥 × (𝑥² + 2𝑥 + 1) + 4 × (𝑥² + 2𝑥 + 1) = 𝑥³ + 2𝑥² + 𝑥 + 4𝑥² + 8𝑥 + 4

Always remember to combine like terms!

(𝑥 + 4) × (𝑥² + 2𝑥 + 1) = 𝑥 × (𝑥² + 2𝑥 + 1) + 4 × (𝑥² + 2𝑥 + 1) = 𝑥³ + 2𝑥² + 𝑥 + 4𝑥² + 8𝑥 + 4

Example Problems: Multiplying Polynomials

Which expression is equivalent to (ℎ² + 9ℎ − 1)(−4ℎ + 3)? (STAAR Algebra 1 May 2018 Question 36)

F. −4ℎ³ − 33ℎ² + 31ℎ − 3

G. 4ℎ³ + 39ℎ² − 23ℎ − 3

H. −4ℎ³ − 39ℎ² + 23ℎ + 3

J. 4ℎ³ + 33ℎ² − 31ℎ + 3

Answer:

(ℎ² + 9ℎ − 1)(−4ℎ + 3)

= ℎ² × (−4ℎ) + 3ℎ² + 9ℎ × (−4ℎ) + 27ℎ + 4ℎ − 3  

= −4ℎ³ + 3ℎ² − 36ℎ² + 27ℎ + 4ℎ − 3  

= −4ℎ³ − 33ℎ² + 31ℎ − 3

The answer is F.

Summary: Multiplying Polynomials

• When multiplying polynomials, the coefficient should be multiplied with a coefficient and the variable should be multiplied with a variable;
• Remove the parenthesis by distributive property;
• Simplify the exponent by rules;
• Don’t forget combine the like terms.

For more examples and applications of Polynomial Multiplication, check this page.

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More related articles

• How to Divide a Polynomial by a Monomial in 2 Simple Steps

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