Guides / Factoring

Special factoring: difference of squares, sum and difference of cubes

Three patterns that turn a two-term expression into factors in one line, once you spot them. The mistakes come from patterns that look similar but aren't: a sum of squares, a cube of a binomial, and signs copied from the wrong place.

Why it works

Each pattern is a multiplication with its middle terms cancelling. Multiply (a − b)(a + b) and you get a² + ab − ab − b²: the ab terms cancel, leaving a² − b². The cube patterns do the same thing with more terms:

1. (a − b)(a² + ab + b²)The difference of cubes pattern.
2. a³ + a²b + ab² − a²b − ab² − b³a times each term, then −b times each term.
3. a³ − b³The a²b terms cancel, and so do the ab² terms. Every sign in the trinomial is chosen to make that happen.

That's why the signs aren't optional. Change one and the cancelling stops, and you get extra terms.

To use a pattern, you have to see each term as a square or a cube. Learn these by sight: squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100; cubes 1, 8, 27, 64, 125, 216. A variable is a square when its exponent is even and a cube when its exponent is a multiple of 3, so x⁶ = (x³)² = (x²)³ is both. Coefficients come along: 9x² = (3x)² and 8x³ = (2x)³.

Example 1: difference of squares with a coefficient

Factor 9x² − 25.

1. (3x)² − 5²9x² is (3x)² and 25 is 5². Two squares with a minus between them.
2. (3x − 5)(3x + 5)a = 3x, b = 5. Check: 9x² + 15x − 15x − 25 = 9x² − 25.

Example 2: sum of cubes

Factor 8x³ + 27.

1. (2x)³ + 3³8x³ is (2x)³ and 27 is 3³. So a = 2x, b = 3.
2. (2x + 3)(a² − ab + b²)Same sign as the original (+) in the binomial; opposite sign (−) on the middle term.
3. (2x + 3)(4x² − 6x + 9)a² = 4x², ab = 6x, b² = 9. The middle term is ab, not 2ab.

The trinomial 4x² − 6x + 9 is finished. It looks like it might factor, but it doesn't over the real numbers (its discriminant is 36 − 144 = −108, which is negative). That's true of the trinomial from every sum or difference of cubes.

Example 3: take out the GCF first

Factor 2x³ − 54. Neither 2 nor 54 is a perfect cube, so the pattern is hidden until the common factor comes out.

1. 2(x³ − 27)Both terms are divisible by 2. Now 27 = 3³ is visible.
2. 2(x³ − 3³)Difference of cubes, with a = x and b = 3.
3. 2(x − 3)(x² + 3x + 9)SOAP: minus in the binomial (same), plus in the middle (opposite), plus on the 9 (always positive).

Example 4: factor until nothing factors: x⁴ − 16

1. (x²)² − 4²x⁴ is (x²)² and 16 is 4².
2. (x² − 4)(x² + 4)Difference of squares. Now check each factor again.
3. (x − 2)(x + 2)(x² + 4)x² − 4 is another difference of squares. x² + 4 is a sum of squares, so it stays.

Check by multiplying: (x − 2)(x + 2) = x² − 4, and (x² − 4)(x² + 4) = x⁴ − 16.

Common mistakes, and the exact line they happen on

1. Factoring a sum of squares

1. x² + 9A sum, not a difference.
2. (x + 3)²This is the mistake. (x + 3)² = x² + 6x + 9. The 6x has nowhere to go.

The difference-of-squares pattern works because +ab and −ab cancel. With a plus between the squares there's nothing to cancel, so x² + 9 doesn't factor over the real numbers. Plug in x = 1 to see it: 1 + 9 = 10, but (1 + 3)² = 16. A common factor can still come out, as in 4x² + 36 = 4(x² + 9), but that's as far as it goes.

2. Getting the cube signs wrong, or factoring the trinomial further

1. x³ − 8 = x³ − 2³Difference of cubes, a = x, b = 2.
2. (x − 2)(x² − 2x + 4)This is the mistake. The minus was copied into the middle of the trinomial. It multiplies out to x³ − 4x² + 8x − 8.

SOAP says the middle sign is the opposite of the original: (x − 2)(x² + 2x + 4). A second version of this slip writes 2ab in the middle, as if the trinomial were a perfect square: (x − 2)(x² + 4x + 4) multiplies out to x³ + 2x² − 4x − 8. The middle term is ab.

The same instinct causes a third slip, on a correct factoring:

1. x³ + 8 = (x + 2)(x² − 2x + 4)Correct.
2. (x + 2)(x − 2)²This is the mistake. x² − 2x + 4 was treated as (x − 2)², which is x² − 4x + 4. The product is x³ − 2x² − 4x + 8, not x³ + 8.

The trinomial from a sum or difference of cubes never factors over the real numbers. Its discriminant is always negative (here 4 − 16 = −12). Once you've applied the cube pattern, the trinomial is done.

3. Missing the GCF

1. 3x² − 4848 isn't a perfect square, and neither is 3x².
2. prime (doesn't factor)This is the mistake. Both terms share a factor of 3.

Take the 3 out first: 3(x² − 16) = 3(x − 4)(x + 4). If the coefficients don't look like squares or cubes, check for a common factor before deciding the pattern doesn't apply.

4. Stopping after one step on x⁴ − 16

1. x⁴ − 16 = (x²)² − 4²Difference of squares.
2. (x² − 4)(x² + 4)This is the incomplete line. It's equal to x⁴ − 16, but x² − 4 still factors.

“Factor completely” means every factor gets checked again. The fully factored answer is (x − 2)(x + 2)(x² + 4). If you are solving x⁴ − 16 = 0, stopping early hides the solutions x = 2 and x = −2 inside a factor you never finished.

5. Confusing a³ − b³ with (a − b)³

1. x³ − 27Difference of cubes.
2. (x − 3)³This is the mistake. (x − 3)³ = x³ − 9x² + 27x − 27. Cubing a binomial produces middle terms.

Test it with x = 1: the original is 1 − 27 = −26, but (1 − 3)³ = −8. The correct factoring is (x − 3)(x² + 3x + 9). It's the same slip as writing (a + b)² = a² + b², run in reverse.

Practice

Factor completely. If something doesn't factor beyond a common factor, say so.

1. 16x² − 81y²

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   (4x)² − (9y)² = (4x − 9y)(4x + 9y).

2. x³ + 125

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   x³ + 5³ = (x + 5)(x² − 5x + 25). The trinomial doesn't factor further.

3. 5x³ − 40

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   GCF first: 5(x³ − 8) = 5(x − 2)(x² + 2x + 4).

4. 81x⁴ − 1

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   (9x² − 1)(9x² + 1), and 9x² − 1 is another difference of squares: (3x − 1)(3x + 1)(9x² + 1).

5. 3x² + 12

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   3(x² + 4), and that's as far as it goes. x² + 4 is a sum of squares, which doesn't factor over the real numbers.