BCCC ASC Rev. 6/2019
Factor the Sum and Difference of Two Cubes
1. Formulas for factoring the Sum and Difference of two cubes:
Sum: a³+b³= (a+b) (a²-ab+b²)
Difference: a³-b³= (a-b) (a²+ab+b²)
Note: Keep in mind that the middle of the trinomial is always opposite the sign of the binomial
2. Identification of Sum and Difference in the given problem:
a³+b³ or a³-b³
↓ ↓
Ex: x³+8 27x³-8
↓ ↓
x³ + 2³ (3x)³ - 2³
↓ ↓ ↓ ↓
let:
a=x b=2 a=3x b=2
(The cubed roots of each term in the original)
Sample of perfect cubes:
1 x
3
27x
3
8 x
3
y
3
8x
3
27 x
6
64x
3
y
3
64 x
9
125x
6
y
3
125 The exponents must be divisible by 3 for a perfect cube
3. Match it to the sum or difference formulas:
Use your “a” and “b” values to match “a” and “b” in the formula you have chosen:
Factor: x
3
+ 8
Sum: a³+b³ = (a+b) (a²-ab+b²)
↑ ↑ ↑ ↑ ↑ ↑ ↑
(cube roots x 2) (x+2) (x²-2x+2²)
So: x
3
+8 = x
3
+2
3
= (x+2) (x²-2x+4)
Note: the middle sign of the trinomial is opposite of the binomial
3. To prove your answer is right multiply (x+2)(x²-2x+4) → using the distributive property :
↔
(x+2)(x²-2x+4)
↔
So: x³-2x²+4x+2x²-4x+8 Simplify by canceling like terms
You get x³+8 which proves that your answer is correct.
