Binomial cube: (a+b)³ = a³ + 3a²b + 3ab² + b³

The two formulas

$(a + b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3$

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

Coefficients 1, 3, 3, 1 (row 4 of Pascal’s triangle).

How to remember

• Powers of $a$: 3, 2, 1, 0 (decreasing).
• Powers of $b$: 0, 1, 2, 3 (increasing).
• Coefficients: 1, 3, 3, 1.
• For $(a-b)^3$: signs alternate + − + −.

Example: $(x + 2)^3$

$= x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 4 + 8$ $= x^3 + 6 x^2 + 12 x + 8$

Example: $(x - 1)^3$

$= x^3 - 3 x^2 + 3 x - 1$

Reverse factoring

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

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

Useful for derivatives and computations

To get $(x + 1)^3$ in a problem, expand directly instead of multiplying thrice.

Common mistakes

1. Missing middle terms ($3 a^2 b$ and $3 a b^2$).
2. Coefficient 2 instead of 3 (confusion with square).
3. Writing $(a+b)^3 = a^3 + b^3$ (WRONG).
4. Signs of $(a-b)^3$: + − + −, not all −.

?À toi de jouer

Expand (2x − 1)³?

A.8x³ − 12x² + 6x − 1B.8x³ − 1C.8x³ − 6x + 1D.6x³ − 12x² + 6x − 1

Valider