1. Factoring the Sum & Difference
of Two Cubes
p.368-371

2. This is a piece of cake, if you have perfect cubes.
What are perfect cubes?

3. This is a piece of cake, if you have perfect cubes.
What are perfect cubes?
Something times something times something.
Where the something is a factor 3 times.
8 is 2 × 2 × 2, so 8 is a perfect cube.
x6 is x2 × x2 × x2 so x6 is a perfect cube.
It is easy to see if a variable is a perfect cube, how?

4. This is a piece of cake, if you have perfect cubes.
What are perfect cubes?
Something times something times something.
Where the something is a factor 3 times.
8 is 2 × 2 × 2, so 8 is a perfect cube.
x6 is x2 × x2 × x2 so x6 is a perfect cube.
It is easy to see if a variable is a perfect cube, how?
See if the exponent is divisible by 3. It’s harder for
integers.

5. The sum or difference of two cubes will factor into a
binomial × trinomial.
(
a + b = ( a + b ) a − ab + b
3
3
2
2
)
same sign
always +
always opposite
(
a − b = ( a − b ) a + ab + b
3
3
same sign
always opposite
2
2
)
always +

6. Now we know how to get the signs, let’s work on
what goes inside.
Square this term to get this term.
(
a + b = ( a + b ) a − ab + b
3
3
2
2
)
Cube root of 1st term
Cube root of 2nd term
Product of cube root of 1st term
and cube root of 2nd term.

7. Try one.
27 x −125 =
3
Make a binomial and a trinomial
with the correct signs.

8. Try one.
27 x −125 =
3
(
−
Cube root of 1st term
)(
+
+
)
Cube root of 2nd term

9. Try one.
27 x −125 = ( 3x − 5)(
3
+
+
)
Square this term to get this term.

10. Try one.
27 x −125 = ( 3x − 5) ( 9 x 2 +
3
+
)
Multiply 3x an 5 to get this term.

11. Try one.
27 x −125 = ( 3x − 5) ( 9 x + 15 x +
3
2
Square this term to get this term.
)

12. Try one.
(
27 x −125 = ( 3 x − 5) 9 x + 15 x + 25
3
2
)
You did it!
Don’t forget the first rule of factoring is to look
for the greatest common factor.