How to Multiply Binomials? (+FREE Worksheet!)

Multiply binomials by distributing each term and combining like terms.

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Multiply Binomials: what to notice and how to work it

Polynomials skill

Polynomial problems reward structure. Before expanding, look for degree, leading term, common factors, and familiar products.

What to notice first

Put the polynomial in standard form when possible. The leading term tells end behavior, and factors reveal zeros.

Common student mistake

Do not cancel or combine unlike terms. \(x^2\), \(x\), and constants are different kinds of terms.

Key formulas and cues

\(a^2-b^2=(a-b)(a+b)\)

\((a+b)^2=a^2+2ab+b^2\)

\(P(c)=0\Rightarrow (x-c)\text{ is a factor}\)

A reliable path

Organize by degreeWrite terms from highest power to lowest power.
Look for structureTry GCF, special products, grouping, or division depending on the expression.
Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.

Worked examples

Combine like terms

Example: \(3x^2+5x-x^2+2x\)

Group x squared terms.
Group x terms.
Combine each group.

Answer: \(2x^2+7x\)

Factor a difference of squares

Example: \(x^2-25\)

Recognize a squared term minus a squared term.
Use a^2 – b^2.
Write conjugate factors.

Answer: \((x-5)(x+5)\)

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Try one before moving on

Try: Factor \(x^2+7x+12\).

Answer: \((x+3)(x+4)\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

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Multiplying binomials is one of the most important skills in Algebra 1 and shows up constantly in factoring, solving equations, and graphing parabolas. The FOIL method, First, Outer, Inner, Last, gives you a reliable shortcut for multiplying any two binomials. This guide covers FOIL step by step, handles special cases like difference of squares and perfect squares, and includes two video lessons with practice problems.

What Is a Binomial?

A binomial is a polynomial with exactly two terms connected by addition or subtraction. Examples: \(\color{blue}{(x + 3)}\), \(\color{blue}{(2x – 5)}\), \(\color{blue}{(a + b)}\). Multiplying two binomials produces a trinomial (three terms) in most cases.

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The FOIL Method

FOIL is a mnemonic for the four pairs you multiply when expanding \(\color{blue}{(a + b)(c + d)}\):

First: multiply the first terms: \(\color{blue}{a \cdot c}\)
Outer: multiply the outer terms: \(\color{blue}{a \cdot d}\)
Inner: multiply the inner terms: \(\color{blue}{b \cdot c}\)
Last: multiply the last terms: \(\color{blue}{b \cdot d}\)

Then combine any like terms (usually the Outer and Inner products).

FOIL Example

\(\color{blue}{(x + 3)(x + 5)}\)

F: \(\color{blue}{x \cdot x = x^{2}}\)
O: \(\color{blue}{x \cdot 5 = 5x}\)
I: \(\color{blue}{3 \cdot x = 3x}\)
L: \(\color{blue}{3 \cdot 5 = 15}\)

Combine: \(\color{blue}{x^{2} + 5x + 3x + 15 = x^{2} + 8x + 15}\)

Special Products

Difference of Squares

\(\color{blue}{(a + b)(a – b) = a^{2} – b^{2}}\)

\(\color{blue}{(x + 5)(x – 5) = x^{2} – 25}\)

Perfect Square Trinomial

\(\color{blue}{(a + b)^{2} = a^{2} + 2\text{ ab } + b^{2}}\) and \(\color{blue}{(a – b)^{2} = a^{2}}\) – \(\color{blue}{2\text{ ab } + b^{2}}\)

\(\color{blue}{(3x – 2)^{2} = 9x^{2} – 12x + 4}\)

Step-by-Step Summary

Identify the two binomials.
Apply FOIL: F, O, I, L products.
Combine the Outer and Inner (like) terms.
Write the result in descending order.
Check for special product patterns to save steps.

Watch: Multiplying Binomials (Video Lesson)

Khan Academy explains the distributive approach and FOIL with multiple examples:

Multiplying Binomials — Worked Examples

Example 1: Multiply \(\color{blue}{(x + 3)(x + 5)}\).

F: \(\color{blue}{x^{2}}\) | O: \(\color{blue}{5x}\) | I: \(\color{blue}{3x}\) | L: \(\color{blue}{15}\)
Combine: \(\color{blue}{x^{2} + 8x + 15}\)

Example 2: Multiply \(\color{blue}{(x – 4)(x + 2)}\).

F: \(\color{blue}{x^{2}}\) | O: \(\color{blue}{2x}\) | I: \(\color{blue}{-4x}\) | L: \(\color{blue}{-8}\)
Combine: \(\color{blue}{x^{2} + (2x – 4x) – 8 = x^{2} – 2x – 8}\)

Example 3: Multiply \(\color{blue}{(2x + 3)(3x – 1)}\).

F: \(\color{blue}{6x^{2}}\) | O: \(\color{blue}{-2x}\) | I: \(\color{blue}{9x}\) | L: \(\color{blue}{-3}\)
Combine: \(\color{blue}{6x^{2} + 7x – 3}\)

Example 4: Multiply \(\color{blue}{(x + 5)(x – 5)}\).

Difference of squares pattern: \(\color{blue}{(x + 5)(x – 5) = x^{2} – 5^{2} = x^{2} – 25}\)

Example 5: Expand \(\color{blue}{(3x – 2)^{2}}\).

Perfect square pattern: \(\color{blue}{(3x)^{2} + 2(3x)(-2) + (-2)^{2} = 9x^{2} – 12x + 4}\)

More Practice: Step-by-Step Video

This Khan Academy video demonstrates the binomial multiplication process in detail:

Exercises for Multiplying Binomials

Expand each product and simplify.

\(\color{blue}{(x + 4)(x + 6)}\)
\(\color{blue}{(x – 3)(x + 7)}\)
\(\color{blue}{(2x + 1)(x – 5)}\)
\(\color{blue}{(3x – 4)(2x + 3)}\)
\(\color{blue}{(x + 2)^{2}}\)
\(\color{blue}{(x – 6)(x + 6)}\)

Answers

\(\color{blue}{x^{2} + 10x + 24}\)
\(\color{blue}{x^{2} + 4x – 21}\)
\(\color{blue}{2x^{2} – 9x – 5}\)
\(\color{blue}{6x^{2} + x – 12}\)
\(\color{blue}{x^{2} + 4x + 4}\)
\(\color{blue}{x^{2} – 36}\)

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Frequently Asked Questions

How do you multiply binomials using FOIL?

FOIL stands for First, Outer, Inner, Last, the four products you need. For (x + 3)(x + 5): First x × x = x², Outer x × 5 = 5x, Inner 3 × x = 3x, Last 3 × 5 = 15. Add them and combine like terms: x² + 8x + 15. FOIL only works for two binomials; bigger products need full distribution.

What is the formula for multiplying binomials?

The general pattern is (a + b)(c + d) = ac + ad + bc + bd. Two special products are worth memorizing: (a + b)² = a² + 2ab + b² and (a + b)(a − b) = a² − b². For example, (x + 4)² = x² + 8x + 16, not x² + 16, and (x + 6)(x − 6) = x² − 36.

How do you multiply binomials using the box method?

Draw a 2-by-2 grid, write one binomial across the top and the other down the side, then fill each box with a product. For (2x − 1)(x + 4), the boxes are 2x², 8x, −x, and −4. Add them and combine the middle terms: 2x² + 7x − 4. The box keeps signs visible, which helps students who drop negatives.

How do you multiply binomials and trinomials?

Distribute each term of the binomial across all three terms of the trinomial, giving six products. For (x + 2)(x² + 3x − 1): x times the trinomial is x³ + 3x² − x, and 2 times it is 2x² + 6x − 2. Combining like terms gives x³ + 5x² + 5x − 2.

What grade is multiplying binomials?

Multiplying binomials is usually taught in Algebra 1, which many students take in grade 8 or 9. It builds on the distributive property from middle school and leads straight into factoring quadratics, since factoring x² + 7x + 12 into (x + 3)(x + 4) is the same skill in reverse.

How do you multiply binomials with square roots?

Use FOIL as usual, then simplify the roots and combine like radicals. For (2 + √3)(1 + √3): 2 × 1 = 2, 2 × √3 = 2√3, √3 × 1 = √3, and √3 × √3 = 3. Adding gives 5 + 3√3. Conjugates are a handy case: (√5 + 2)(√5 − 2) = 5 − 4 = 1.

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How to Multiply Binomials? (+FREE Worksheet!)