Identity

Sum and difference of two cubes

Algebra • Expressions • Algebraic Identities

Formula

a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp ab + b^{2})

For all

a

and

b

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

and

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

How to use it

1Read the signs off the rule: the linear factor copies the sign, and the middle term of the quadratic takes the opposite one.
2Use it to cancel: $\frac{a ^{2} + ab + b ^{2}}{a ^{3} - b ^{3}}$ collapses to $\frac{1}{a - b}$ in one move.
3Use it for divisibility: $a + b$ always divides $a^{3} + b^{3}$ , which turns a search over integer pairs into a search over divisors.

Factorise

27 x^{3} - 8

Step-by-step

1
$27 x^{3} - 8 = (3 x)^{3} - 2^{3}$
Why: Write both terms as cubes so the identity applies directly.
2
$= (3 x - 2) (9 x^{2} + 6 x + 4)$

Answer

(3 x - 2) (9 x^{2} + 6 x + 4)

Watch out

The quadratic factor is

a^{2} \mp ab + b^{2}

and not

(a \mp b)^{2}

: the middle coefficient is

1

, not

2

. Over the reals that quadratic never factorises further unless

a

and

b

are equal.

Also known as

sum of cubes, difference of cubes, a cubed plus b cubed, cube factorisation