Factoring the Difference of Two Cubes (a3 - b3)

Factoring the Difference of Two Cubes

How to solve factoring the difference of two cubes problems: formula, examples, and their solutions.

Formula

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

a3 - b3 = (a - b)(a2 + ab + b2)

See the blue colored signs.

If the sign of b3 term in (a3 - b3) is (-),

then the sign of b term in (a - b) is (-)
and the sign of ab term in (a2 + ab + b2) is (+).

Proof

Factoring the Difference of Two Cubes: Proof of the Formula

Solve (a + b)(a2 - ab + b2).

Multiplying polynomials

Then a2b terms and ab2 terms are cancelled.

So (a - b)(a2 + ab + b2) = a3 - b3.

Switch both sides.

Then a3 - b3 = (a - b)(a2 + ab + b2).

Example 1

Factor the given polynomial. x^3 - 125

x3 - 125 = x3 - 53
= (x - 5)(x2 + x⋅5 + 52)

Example 2

Factor the given polynomial. 8a^3 - 27

8a3 - 27 = (2a)3 - 33
= (2a - 3)((2a)2 + 2a⋅3 + 32)

Example 3

Factor the given polynomial. p^6 - 1

First change it as two squares and factor it.

p6 - 1 = (p3)2 - 12
= (p3 + 1)(p3 - 1)

Factoring the diffeerence of squares

Factor the sum and difference of two cubes.

p3 + 13 = (p + 1)(p2 - p⋅1 + 12)

Factoring the sum of two cubes

p3 - 13 = (p - 1)(p2 + p⋅1 + 12)

When factoring 6th powers,
first change it as two squares and factor it,
then factor the sum of two cubes.

If you change this order,
it'll be quite tough to solve it.
(See the next image.)

Example 4

Factor the given polynomial. p^6 - 1

This solution is to show you
how complex it is
if you don't follow the order:
'square → cube'

First change it as two cubes and factor it.

p6 - 1 = (p2)3 - 13
= (p2 - 1)((p2)2 + p2⋅1 + 12)

To factor ((p2)2 + p2⋅1 + 12),
complete the square.

Change +p2 to +2p2
and write -p2 to undo the change.

Completing the square

Factor the differences of two squares.

p2 - 1 = (p + 1)(p - 1)

(p2 + 1)2 - p2 = (p2 + 1 + p)(p2 + 1 - p)

Factoring the diffeerence of squares