Algebra › Factoring

Factoring

Factoring is multiplication in reverse. You take an expression and pull it apart into smaller pieces that multiply to give it back.

Factor a quadratic, step by step

Find two numbers that add to the middle and multiply to the end.

Start with

x² + 5x + 6

Find two numbers that multiply to 6 and add to 5

Those numbers are 2 and 3

Write the factors

Step 0 of 3

Why factor at all?

A factored form makes things easy:

Roots become obvious — set each factor to zero.
Cancellation in fractions only works on factors, not on terms.
Many integrals and limit problems clear up after one factor.

Step 1 — pull out a common factor

Always look for this first. Example:

6x³ + 9x² = 3x² (2x + 3)

Both terms share a 3 and at least an x² — pull them out front and bag the rest in brackets.

Step 2 — recognise a special pattern

Difference of squares — a² − b² = (a + b)(a − b)
Perfect square — a² + 2ab + b² = (a + b)²
Difference of cubes — a³ − b³ = (a − b)(a² + ab + b²)

Step 3 — split the middle (for x² + bx + c)

Find two numbers that multiply to c and add to b. Those are the numbers inside the brackets.

x² + 5x + 6 → 2 × 3 = 6 and 2 + 3 = 5 → (x + 2)(x + 3)

Sign rules

If c is positive, both numbers share a sign (both + if b is +, both − if b is −). If c is negative, the numbers have opposite signs and the bigger one matches the sign of b.

Quick check

Factor 4x + 12.
Factor x² − 9.
Factor x² − 7x + 10.

Answers: 4(x + 3), (x + 3)(x − 3), and (x − 2)(x − 5).

Quick check

Factor x² − 9.

Quick check

Factor x² − 7x + 10.