Factoring Special Cases

Two patterns are worth recognising on sight.

Difference of squares: two perfect squares with a minus between them.
x² − 49 = (x + 7)(x − 7)
16x² − 25 = (4x + 5)(4x − 5)

A *sum* of squares like x² + 49 does not factor over the real numbers. That difference matters.

Perfect square trinomial: first and last terms are squares, middle is twice the product of their roots.
x² + 12x + 36 = (x + 6)², since 2·6 = 12. ✓

Always remove a GCF first. 8x² − 32 is not obviously a difference of squares until it becomes 8(x² − 4) = 8(x + 2)(x − 2).

Difference of squares, a² − b² = (a + b)(a − b), holds for any expressions a and b, including ones already containing radicals or higher powers: x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2). Recognising that a factorisation can repeat is what "factor completely" demands.

A sum of squares a² + b² is irreducible over ℝ but factors over ℂ as (a + bi)(a − bi), which is why the answer to "does it factor" depends on the number system in play — a distinction that becomes explicit in Algebra 2.

Perfect square trinomials satisfy a² ± 2ab + b² = (a ± b)². Verifying the middle term is essential: x² + 13x + 36 has square first and last terms but is not a perfect square, since 2·6 = 12 ≠ 13. It factors instead as (x + 4)(x + 9).

The perfect square pattern is also the engine of completing the square, where a constant is manufactured to force the pattern to hold.