The Sum and Difference of Cubes

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A cube identity handles two terms whose exponents and coefficients are perfect cubes. Unlike the square identity, both the sum and the difference of cubes factor over the reals.

Visual guide

VISUAL GUIDECube identities have different real zeros

x³ − 1
x³ + 1

For b = 1, x³ − 1 crosses zero at x = 1, while x³ + 1 crosses at x = −1. The quadratic cofactors stay positive on the real line, so each cubic has only that one real zero.

Method

A^3-B^3=(A-B)(A^2+AB+B^2).

A^3+B^3=(A+B)(A^2-AB+B^2).

The sign in the first factor matches the original; the middle sign in the quadratic is opposite.

Worked example

Since $8x^3+27=(2x)^3+3^3$ , its factors are

(2x+3)(4x^2-6x+9).

Multiplying gives $8x^3-12x^2+18x+12x^2-18x+27$ : the middle terms cancel. This is why changing the quadratic's middle sign breaks the identity.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which factorization equals

x^3+8

(x+2)(x²−2x+4)

(x+2)(x²+2x+4)

(x+2)³

Hint 1 · Find a starting point

8 is 2³; a sum of cubes differs from a cube of a sum.

Hint 2 · Take the next step

Use a³+b³ = (a+b)(a²−ab+b²).

Show the reasoning

Answer: (x+2)(x²−2x+4)

The alternating middle sign cancels the unwanted x² and x terms.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

A sum of cubes is not a cube of a sum: $(A+B)^3$ also contains two mixed terms.

Check your understanding

Factor $x^3-64$ .

Show answer

$(x-4)(x^2+4x+16)$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →