Difference of Squares & Sum/Difference of Cubes

Three special factoring formulas

These three formulas should be memorized — they appear constantly on exams:

Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Step-by-step: Factor $x^3 - 27$

Step 1 — Recognize the pattern. $x^3 - 27 = x^3 - 3^3$. This is a difference of cubes with $a = x$ and $b = 3$.

Step 2 — Apply the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$:

$x^3 - 27 = (x - 3)(x^2 + 3x + 9)$

Step 3 — Check by expanding:

$(x - 3)(x^2 + 3x + 9) = x^3 + 3x^2 + 9x - 3x^2 - 9x - 27 = x^3 - 27 \checkmark$

Step 4 — Can we factor further? Check $x^2 + 3x + 9$: discriminant $= 9 - 36 = -27 < 0$. No real roots, so this factor is irreducible over the reals.

More examples

Difference of squares: $4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)$

Sum of cubes: $8x^3 + 125 = (2x)^3 + 5^3 = (2x + 5)(4x^2 - 10x + 25)$

Memory trick for cubes

The factored form is always: (binomial)(trinomial). The binomial has the same sign as the original. The trinomial follows the pattern: "Square, Opposite Product, Square" — $a^2$, $\mp ab$, $b^2$ (the middle sign is opposite to the binomial's sign).

Common mistakes

• Confusing difference and sum. $a^2 - b^2$ factors, but $a^2 + b^2$ does NOT factor over the reals.
• Forgetting to check if $b$ is a perfect cube/square. $x^3 - 12$ is NOT a difference of cubes because 12 isn't a perfect cube.
• Sign errors in the trinomial. For $a^3 - b^3$: binomial is $(a - b)$, trinomial is $(a^2 + ab + b^2)$ — all positive inside.

Try it in the visualization

Select difference of squares, difference of cubes, or sum of cubes. Adjust $a$ and $b$ values. The factored form updates live, the graph confirms roots, and the expansion check verifies the answer.