Factoring

Factor any expression using the four methods in order: highest common factor, difference of squares, quadratic trinomials (monic and non-monic) and grouping.

The factoring checklist

Factoring is expanding in reverse. With so many methods it helps to try them in a fixed order every time:

Example 1: HCF then difference of squares●

Factor $3x^3 - 12x$.

$= 3x(x^2 - 4)$

3 and x divide every term.

$= 3x(x - 2)(x + 2)$

x² − 4 = x² − 2² is a difference of squares.

Monic quadratics: $x^2 + bx + c$

Find two numbers that multiply to $c$ and add to $b$. Then $x^2 + bx + c = (x + m)(x + n)$.

Example 2●

Factor $x^2 - x - 12$.

Numbers: $3$ and $-4$.

Pairs that multiply to −12: (1, −12), (2, −6), (3, −4), (−3, 4)... Only 3 and −4 add to −1.

$x^2 - x - 12 = (x + 3)(x - 4)$

Non-monic quadratics: $ax^2 + bx + c$

When $a \ne 1$, find two numbers that multiply to $ac$ and add to $b$. Use them to split the middle term, then group.

Example 3●●

Factor $6x^2 + x - 2$.

We need two numbers multiplying to $-12$ and adding to $1$: they are $4$ and $-3$.

a × c = 6 × (−2) = −12.

$6x^2 + 4x - 3x - 2$

Split the middle term x into 4x − 3x.

$= 2x(3x + 2) - 1(3x + 2)$

Take a common factor from each pair. The brackets must match.

$= (3x + 2)(2x - 1)$

Grouping in pairs

Example 4●●

Factor $x^3 + 2x^2 - 9x - 18$.

$= x^2(x + 2) - 9(x + 2)$

Take a common factor out of the first pair and out of the second pair.

$= (x + 2)(x^2 - 9)$

$= (x + 2)(x - 3)(x + 3)$

Check every factor: x² − 9 is a difference of squares.

Flashcards

First step in any factoring

Take out the highest common factor.

Monic x² + bx + c

Two numbers that multiply to $c$ and add to $b$.

Non-monic ax² + bx + c

Two numbers that multiply to $ac$ and add to $b$, then split and group.

Four terms

Try grouping in pairs.

A² − B²

$(A - B)(A + B)$

Practice

Factor $x^2 - 49$.
Factor $2x^2 + 7x + 3$.
Which is the first step to factor $5x^2 - 20$?
$9x^2 + 12x + 4 =$
$x^2 + 2x - 15 = (x + 5)(x + k)$. Find $k$.
Factor $3x^2 - 10x - 8$.
Factor $ab + 3a + 2b + 6$.
Factor fully: $2x^2 - 18 = 2(x - 3)(x + k)$. Find $k$.
Which of these is a perfect square?