Binomial Expansion Calculator – Expand & Simplify Binomials

Expand and simplify binomial expressions instantly with our free online binomial expansion calculator. Handles products, squares, and cubes of binomials with full step-by-step solutions using FOIL and special product formulas.

Operation Type

First Term (a)

Second Term (b)

Examples:

Understanding Binomial Expansion

A binomial is an algebraic expression with two terms, like (x + 3) or (2a - b). Expanding binomials means multiplying them out and simplifying. This skill is fundamental in algebra and appears everywhere from solving equations to calculus.

Special patterns make expansion easier. The square of a binomial (a + b)² always equals a² + 2ab + b². The cube (a + b)³ follows the pattern a³ + 3a²b + 3ab² + b³. The product (a + b)(a - b) always simplifies to a² - b², called the difference of squares.

Binomial Expansion Formulas

Square of a Binomial

(a + b)² = a² + 2ab + b²

First squared, plus twice the product, plus second squared.

Cube of a Binomial

(a + b)³ = a³ + 3a²b + 3ab² + b³

Coefficients follow Pascal's triangle: 1, 3, 3, 1.

FOIL Method

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

First, Outer, Inner, Last – multiply each pair of terms.

Difference of Squares

(a + b)(a - b) = a² - b²

Sum times difference equals difference of squares.

Worked Examples

Example 1: (x + 3)²

Using (a + b)² = a² + 2ab + b²

a = x, b = 3

x² + 2(x)(3) + 3²

x² + 6x + 9

The middle term is twice the product: 2 × x × 3 = 6x.

Example 2: (2x + 5)²

Using (a + b)² = a² + 2ab + b²

a = 2x, b = 5

(2x)² + 2(2x)(5) + 5²

4x² + 20x + 25

Remember to square the coefficient: (2x)² = 4x².

Example 3: (x + 2)³

Using (a + b)³ = a³ + 3a²b + 3ab² + b³

a = x, b = 2

x³ + 3(x²)(2) + 3(x)(2²) + 2³

x³ + 6x² + 12x + 8

Coefficients 1, 3, 3, 1 come from Pascal's triangle row 3.

Example 4: (x + 3)(x - 3)

Using difference of squares: (a + b)(a - b) = a² - b²

a = x, b = 3

x² - 3²

x² - 9

The middle terms cancel: +3x and -3x add to zero.

Example 5: (2x + 1)(x - 4)

Using FOIL:

First: 2x × x = 2x²

Outer: 2x × (-4) = -8x

Inner: 1 × x = x

Last: 1 × (-4) = -4

Combine: 2x² - 8x + x - 4 = 2x² - 7x - 4

FOIL ensures you multiply every term in the first binomial by every term in the second.

Example 6: (3x)² - (2)²

This is the result of expanding (3x + 2)(3x - 2)

Using difference of squares:

(3x)² - 2²

9x² - 4

The factored form (3x + 2)(3x - 2) expands to 9x² - 4.

Quick Fact

Pascal's Triangle gives binomial coefficients. For (a + b)^n, the coefficients come from row n of Pascal's triangle. Row 0: 1. Row 1: 1, 1. Row 2: 1, 2, 1. Row 3: 1, 3, 3, 1. Row 4: 1, 4, 6, 4, 1. This pattern was known to mathematicians in China, Persia, and Europe centuries before Pascal.

Frequently Asked Questions

What does FOIL stand for?

FOIL is a mnemonic: First, Outer, Inner, Last. It tells you which terms to multiply when expanding two binomials. Multiply the first terms, then outer terms, then inner terms, then last terms, and add them all.

Why is (a + b)² not equal to a² + b²?

Because you're missing the middle term! (a + b)² = (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b². The cross terms (ab and ba) add up to 2ab. This is a common mistake.

Can I expand (a - b)² the same way?

Yes, but watch the signs. (a - b)² = a² - 2ab + b². The middle term is negative because you're multiplying a by -b twice. The last term is still positive because (-b)² = b².

What if there are coefficients?

Treat the coefficient as part of the term. (2x)² = 4x², not 2x². Square both the coefficient and the variable. For (3x + 4)²: (3x)² + 2(3x)(4) + 4² = 9x² + 24x + 16.

How do I expand (a + b + c)²?

For trinomials: (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc. Every pair of terms contributes a cross term. There are three squared terms and three cross terms.

When would I use binomial expansion?

Binomial expansion appears in factoring, solving quadratic equations, completing the square, calculus (derivatives and integrals), probability (binomial distribution), and physics (Taylor series approximations).

Other Free Tools

Search tools