Special Factorizations
Four patterns that factor themselves once you recognize them: difference of squares, perfect-square trinomials, and the sum and difference of cubes. Learn each fingerprint — and the one pattern that never factors.
1 Understand
The core idea in plain language.
What it is. Difference of squares: a2 − b2 = (a + b)(a − b). Perfect-square trinomial: a2 ± 2ab + b2 = (a ± b)2. Sum/difference of cubes: a3 ± b3 = (a ± b)(a2 ∓ ab + b2). Sum of squares a2 + b2 does not factor over the reals.
Why it matters. These are the fingerprints of factored form. Spotting a difference of squares inside a bigger problem collapses it in one step — and knowing sum of squares is prime stops you from “factoring” something unfactorable.
Where it is used. Simplifying rational expressions · solving higher-degree equations · conjugate rationalization.
2 See It
Diagrams that make the idea visual.
Difference of squares, seen
9x2 − 16 is (3x)2 − 42: two perfect squares with a minus between them. The fingerprint says (3x + 4)(3x − 4) — the middle terms cancel on the way back.
The cube patterns
x3 − 8 is x3 − 23: a difference of cubes. The pattern gives a binomial (x − 2) times a trinomial (x2 + 2x + 4). Watch the signs: minus, plus, plus. For a sum of cubes the signs flip: plus, minus, plus.
3 Worked Examples
Follow each step. The pattern is always the same.
- Fingerprint: (3x)2 − 42 — two squares, minus between.
- a² − b² = (a + b)(a − b) with a = 3x, b = 4.
Check: Multiply back: 9x² − 12x + 12x − 16 = 9x² − 16 (checks).
- Fingerprint: x2 + 2·x·5 + 52 — first and last are squares, middle is twice the product.
- Perfect-square trinomial: (x + 5)².
Check: (x + 5)² = x² + 10x + 25 (checks).
- First x2 and last 32 are squares; middle −6x = −2·x·3.
- Perfect square with minus: (x − 3)².
Check: (x − 3)² = x² − 6x + 9 (checks).
- Fingerprint: x3 − 23 — difference of cubes.
- a³ − b³ = (a − b)(a² + ab + b²) = (x − 2)(x² + 2x + 4).
Check: Expanding (x − 2)(x² + 2x + 4) returns x³ − 8 (checks).
- Two squares — but with a PLUS between them.
- Sum of squares is prime over the reals: no (a + b)(a − b) pattern applies.
Check: No real pair (a + b)(a − b) expands to x² + 9.
4 Common Mistakes
These errors show up on almost every quiz. Spot them now and they will never cost you points.
5 Quick Check
Try each one on paper first, then reveal the answer.
Key Points to Remember
- a² − b² = (a + b)(a − b) — two squares with a minus between.
- a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)² — check the middle = 2·product.
- a³ − b³ = (a − b)(a² + ab + b²) — SOAP signs: same, opposite, always plus.
- a³ + b³ = (a + b)(a² − ab + b²).
- a² + b² (sum of squares) is PRIME over the reals — never factor it.
- Always multiply back: the middle terms must cancel (squares) or rebuild (cubes).