Two formulas, and one three-letter word that keeps the signs straight.
Factor
x³ − 27
a³ − b³ = (a − b)(a² + ab + b²) with a = x and b = 3. The short bracket takes the same sign, the long bracket is all plus: (x − 3)(x² + 3x + 9). A is the perfect-square trinomial x² + 6x + 9, a different pattern entirely.
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1000 |
A variable is a cube when its exponent divides by 3 — x³, x⁶, 8x³ = (2x)³.
These formulas only apply to a binomial. Unlike a sum of squares, a sum of cubes does factor — a² + b² does not.
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)
the binomial keeps the original sign
the middle term flips it
the last term never changes
SOAP. It is the only part of these formulas people get wrong.
a = x, b = 3
since 3³ = 27
(x − 3)(x² + 3x + 9)
same, opposite, positive
| Piece | Built from | Here |
|---|---|---|
| a² | x squared | x² |
| ab | x times 3 | 3x |
| b² | 3 squared | 9 |
8x³ = (2x)³, 125 = 5³
a = 2x, b = 5
(2x + 5)(a² − ab + b²)
a sum, so plus then minus
a² = 4x², ab = 10x, b² = 25
square and multiply
(2x + 5)(4x² − 10x + 25)
done
a² with a = 2x is 4x², not 2x². And ab is 2x × 5 = 10x, not 5x. Every coefficient comes along for the ride.
x² + 3x + 9
try the discriminant
9 − 4(1)(9) = −27
negative
no real roots → prime
leave it alone
The second bracket from a cube formula never factors over the real numbers. If yours does, check the arithmetic.
Watch for a common factor first, though: 2x³ − 54 is 2(x³ − 27), and only then does the cube formula apply.
x6 is both (x³)² and (x²)³. Factor completely: x6 − 1. Treat it as a difference of squares first.
x6 − 1 = (x³)² − 1² = (x³ − 1)(x³ + 1)
difference of squares
x³ − 1 = (x − 1)(x² + x + 1)
SOAP, minus version
x³ + 1 = (x + 1)(x² − x + 1)
SOAP, plus version
(x − 1)(x + 1)(x² + x + 1)(x² − x + 1)
four factors
Treating x6 − 1 as (x²)³ − 1 gives (x² − 1)(x4 + x² + 1), and x4 + x² + 1 still factors, which is easy to miss. Squares first lands on all four pieces. A sum of squares such as x² + 1 is never one of them.
Factor 125 − x³.
a = 5, b = x
a is whatever comes first
(5 − x)(25 + 5x + x²)
x = 1: 124 = 4 × 31 ✓
Different: the order is reversed, so the short bracket is (5 − x), not (x − 5). Both can appear in the choices.
Factor x³ − 164.
164 = (14)³
cube root of top and bottom
(x − ¼)(x² + ¼x + 116)
ab = ¼x, b² = 116
Different: b is a fraction, so b² shrinks — 116, not 18. At x = 1 both sides give 6364.
Factor 8a⁶ − b³.
8a⁶ = (2a²)³
6 ÷ 3 = 2 on the exponent
(2a² − b)((2a²)² + 2a²b + b²)
SOAP with a = 2a²
(2a² − b)(4a⁴ + 2a²b + b²)
a = b = 1: 7 = 1 × 7 ✓
Different: the cube root of a⁶ is a², and squaring it in the long bracket gives a⁴.
Squaring the block means squaring the 2 and doubling the exponent. Writing 4a² or 2a⁴ is the slip the wrong choices are built on.
a and b can be any expression. Factor (x − 2)³ + 27 with a = x − 2 and b = 3.
(a + b) = x − 2 + 3 = x + 1
the short bracket
a² − ab + b² = (x − 2)² − 3(x − 2) + 9
the long bracket, block kept whole
= x² − 4x + 4 − 3x + 6 + 9 = x² − 7x + 19
tidy
(x + 1)(x² − 7x + 19)
x = 0: −8 + 27 = 19 = (1)(19) ✓
| Also asked | The move |
|---|---|
| a + b = 5, ab = 6; find a³ + b³ | a² − ab + b² = (a + b)² − 3ab = 25 − 18 = 7, so a³ + b³ = 5 × 7 = 35 |
| a − b = 2, ab = 5; find a³ − b³ | a² + ab + b² = (a − b)² + 3ab = 4 + 15 = 19, so a³ − b³ = 2 × 19 = 38 |
| (47³ − 3³) ÷ (47² + 47 × 3 + 3²) | the bottom is the long bracket of a³ − b³, so the answer is 47 − 3 = 44 |
| x³ + y³ + x + y | (x + y)(x² − xy + y²) + (x + y) = (x + y)(x² − xy + y² + 1) |
Cube items test SOAP directly (“which is a factor of 64x³ − 1”), or hide the pattern: a sixth power, a bracket as a, a fraction built from two cubes, or a³ ± b³ from a ± b and ab. If you see two cubes, write the SOAP brackets at once — and test a choice by putting in a small x such as 0 or 1.
Factor
27x³ − 8
Here a = 3x and b = 2. Squaring all of 3x gives 9x², and the middle term is ab = 6x with a plus: (3x − 2)(9x² + 6x + 4). C is the sum-of-cubes pattern, which belongs to 27x³ + 8.
