1. AA Notes 4.3: Factoring Sums & Differences of Cubes
Objectives • Define “cubes”
• Factor Sums of Cubes
• Factor Difference of Cubes

2. Concept: What are Perfect Cubes?
Something times something times something.
Where the something is a factor 3 times.
Example: 2  2  2 = 8, so 8 is a perfect cube.
x2  x2  x2 = x6 so x6 is a perfect cube.
It is easy to see if a variable is a perfect cube.
Just see if the exponent is divisible by 3.

3. Concept: What Is A Sum or Difference of Cubes
Has two terms
The terms are separated by a + or – sign
Each term is a perfect cube
The sum or difference of two cubes will factor into a binomial  trinomial.

4. Concept: Factoring Sums of Cubes
same sign
always opposite
always +

5. Concept: Factoring Differences of Cubes
same sign
Sum of Cubes
always opposite
always +
Difference of Cubes

6. Concept: Factoring Sums & Differences of Cubes
Now we know how to get the signs, let’s work on
what goes inside.
Square this term to get this term.
Cube root of 1st term
Product of cube root of 1st term and cube root of 2nd term.
Cube root of 2nd term

7. You did it! Concept: Factoring Sums & Differences of Cubes cont… 3x 5
Factor this Difference of Cubes. Watch your signs!!! 3x 5
9x2
15x 25
Cube root of 1st term
Multiply these
to get this.
Square this term
to get this term.
Cube root of 2nd term
Square this term
to get this term.
You did it!
I hope you took notes because we are done here!!!

8. Factoring Sums & Differences of Cubes
Example 2: Factor each of the following polynomials
8x3 – 27
= (2x – 3)((2x)2 + (2x)(3) + (3)2)  Use the pattern
a3 – b3 = (a – b)(a2 + ab + b2)
= (2x – 3)(4x2 + 6x + 9)
b) 9x4 – 9x
= 9x(x3 – 1)  First remove the GCF
= 9x(x – 1)(x2 + x + 1)  Factor the difference of cubes x3 – 1

9. Homework Time