Special Factoring Patterns

Factor difference of squares, perfect square trinomials, and sum/difference of cubes. Math 10-C.

What you'll learn in this lesson

Perfect Square Trinomials. Learn how to recognize and factor perfect square trinomials by checking three conditions, and apply the shortcut to factor expressions like 4x^2 + 20x + 25 as (2x + 5)^2.
Difference of Squares. Discover the difference of squares pattern and use it to factor binomials like 9x^2 - 25 into (3x + 5)(3x - 5). Learn why a sum of squares cannot be factored over the reals.
GCF First, Then Special Patterns. Master the full factoring checklist: always pull out the GCF first, then apply special patterns. Work through multi-step problems including repeated difference of squares.

What Is a Perfect Square Trinomial?. When you square a binomial, you always get a trinomial with a specific structure:

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

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

The first and last terms are perfect squares, and the middle term equals twice the product of the square roots of the first and last terms.

A trinomial with this exact structure is called a perfect square trinomial (PST).

Three Conditions to Check. A trinomial is a perfect square trinomial if ALL three conditions hold:

Condition 1: The first term is a perfect square.

Condition 2: The last term is a perfect square.

Condition 3: The middle term equals 2 × (√(first)) × (√(last)).

If all three conditions are met, the trinomial factors as a squared binomial. The sign in the binomial matches the sign of the middle term.

Example 1: Factor 4x^2 + 20x + 25. Step 1 — Check the first term: Is 4x^2 a perfect square? Yes — (2x)^2 = 4x^2, so a = 2x.

Step 2 — Check the last term: Is 25 a perfect square? Yes — 5^2 = 25, so b = 5.

Step 3 — Check the middle term: Does 2(2x)(5) = 20x? Yes.

Step 4 — Identify the sign: The middle term is positive, so use (a + b)^2.

Answer: 4x^2 + 20x + 25 = (2x + 5)^2

Example 2: Factor 25a^2 - 20a + 4. Step 1 — Check the first term: √(25a^2) = 5a, so the formula value of a is 5a.

Step 2 — Check the last term: √(4) = 2, so b = 2.

Step 3 — Check the middle term: 2(5a)(2) = 20a — matches 20a. ✓

Step 4 — Identify the sign: The middle term is negative, so use (a - b)^2.

Answer: 25a^2 - 20a + 4 = (5a - 2)^2

Example 3: Factor 4a^2 + 4ab + b^2. Step 1 — Check the first term: √(4a^2) = 2a.

Step 2 — Check the last term: √(b^2) = b.

Step 3 — Check the middle term: 2(2a)(b) = 4ab — matches. ✓

Step 4 — Identify the sign: The middle term is positive, so use (a + b)^2.

Answer: 4a^2 + 4ab + b^2 = (2a + b)^2

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