Factoring Polynomials

Special Factoring Patterns

Recognizing and factoring the difference of squares and perfect square trinomials.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Factoring Trinomials

Patterns Worth Recognizing on Sight

Before reading on, expand (x + 5)(x − 5) using FOIL, and notice what happens to the two middle terms. Then expand (x + 4)² and compare its structure to the trinomials factored in the last lesson.

Expanding (x + 5)(x − 5) gives x² − 5x + 5x − 25 — the middle terms are opposites and cancel completely, leaving just x² − 25. This always happens whenever a sum and difference of the same two terms are multiplied, which is why it's worth recognizing as its own pattern rather than working through FOIL every time.

Definition — Difference of Squares

A binomial of the form a² − b² always factors as (a + b)(a − b), since the cross terms in FOIL cancel exactly.

Worked Example — Factoring a Difference of Squares

Factor x² − 49. Recognize this as a difference of squares: a = x and b = 7, since 49 = 7². Factor directly: (x + 7)(x − 7).

Expanding (x + 4)² gives x² + 4x + 4x + 16 = x² + 8x + 16 — the middle term, 8x, is exactly double the product of x and 4, and the last term, 16, is 4². This 'double the product' structure is the second pattern worth recognizing.

Definition — Perfect Square Trinomial

A trinomial of the form a² + 2ab + b² factors as (a + b)², and a² − 2ab + b² factors as (a − b)² — recognizable because the first and last terms are perfect squares and the middle term is exactly twice their product.

Worked Example — Factoring a Perfect Square Trinomial

Factor x² − 10x + 25. Check the pattern: x² and 25 = 5² are both perfect squares, and the middle term, −10x, is exactly −2(x)(5). This matches the perfect square pattern with a subtraction: (x − 5)².

a^{2} - b^{2} = (a+b)(a-b) \qquad a^{2} \pm 2ab + b^{2} = (a \pm b)^{2}

Tip

Before defaulting to the general sum-and-product method from the last lesson, quickly check whether a trinomial fits one of these special patterns — recognizing them on sight is much faster than searching for a number pair every time.

Common Mistakes

Attempting to factor a sum of squares, like x² + 25, as if it followed the difference-of-squares pattern.
The difference-of-squares pattern only applies to subtraction, a² − b² — a sum of two squares, a² + b², doesn't factor using real numbers at all.
Misidentifying a trinomial as a perfect square without checking that the middle term is exactly twice the product of the square roots of the first and last terms.
Verify all three conditions — first term a perfect square, last term a perfect square, and middle term exactly double their product — before applying the perfect-square shortcut.

Key Takeaways

A difference of squares, a² − b², always factors as (a + b)(a − b).
A perfect square trinomial, a² ± 2ab + b², factors as (a ± b)².
Recognizing these patterns on sight is faster than the general sum-and-product method, but only applies when a trinomial genuinely fits one of them.

Summary

Special factoring patterns speed up work on expressions with a recognizable structure. With a complete factoring toolkit built up, the next unit puts it to work graphing and solving quadratic functions.

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