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Special Binomial Products — Definition, Formula & Examples

Special binomial products are three commonly used patterns for multiplying binomials that produce predictable results: the square of a sum, the square of a difference, and the difference of squares. Memorizing these patterns lets you expand or factor expressions quickly without using FOIL every time.

Given real numbers or algebraic expressions aa and bb, the special binomial products are the identities (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2, (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2, and (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. Each identity follows directly from the distributive property of multiplication over addition.

Key Formula

(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)(a−b)=a2−b2\begin{aligned}(a+b)^2 &= a^2 + 2ab + b^2 \\ (a-b)^2 &= a^2 - 2ab + b^2 \\ (a+b)(a-b) &= a^2 - b^2\end{aligned}
Where:
  • aa = The first term in the binomial
  • bb = The second term in the binomial

How It Works

To use a special binomial product, identify which pattern matches your expression. For a squared binomial like (a+b)2(a+b)^2, square the first term, add twice the product of both terms, and square the last term. For conjugates like (a+b)(a−b)(a+b)(a-b), simply write the difference of the squares of each term. These patterns work whether aa and bb are numbers, variables, or more complex expressions.

Worked Example

Problem: Expand (3x+5)2(3x + 5)^2 using the square-of-a-sum pattern.
Identify a and b: Here a=3xa = 3x and b=5b = 5.
Square the first term: Compute a2a^2.
(3x)2=9x2(3x)^2 = 9x^2
Twice the product: Compute 2ab2ab.
2(3x)(5)=30x2(3x)(5) = 30x
Square the last term: Compute b2b^2.
52=255^2 = 25
Answer: (3x+5)2=9x2+30x+25(3x+5)^2 = 9x^2 + 30x + 25

Why It Matters

These patterns appear constantly in Algebra 1, Algebra 2, and precalculus whenever you factor trinomials, complete the square, or simplify rational expressions. In physics and engineering, recognizing a difference of squares helps simplify formulas quickly—such as rewriting vf2−vi2v_f^2 - v_i^2 as (vf+vi)(vf−vi)(v_f + v_i)(v_f - v_i).

Common Mistakes

Mistake: Writing (a+b)2=a2+b2(a+b)^2 = a^2 + b^2 and forgetting the middle term 2ab2ab.
Correction: Squaring a binomial always produces three terms. The middle term 2ab2ab comes from the cross-multiplication and cannot be omitted.