Factoring the Sum and
Difference of Cubes
Essential Question
Compare/contrast sum &
difference of cubes with
difference of squares.
Sum of Two Cubes
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Sum (addition) sign
Two Cubes
Difference of Two Cubes
3
3
2
2
𝑎𝑎 − 𝑏𝑏 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Difference (subtraction) sign
Two Cubes
Use this pattern:
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Write a new binomial without the exponents.
Use the new binomial to create the trinomial.
1. Square the first and last terms of the
binomial to create the first and last terms
of the trinomial.
2. Multiply the terms of the binomial to
create the middle term of the trinomial.
3. Sign of the 2nd term is opposite of the
binomial.
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Factor:
Write as a sum of
2 cubes:
Write the binomial
without the cubes:
3
𝑥𝑥
3
+8
3
= 𝑥𝑥 +2
= (𝑥𝑥 + 2)(
)
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
= (𝑥𝑥 + 2)(
Square the first
and last terms:
Multiply the terms
in the binomial:
Opposite signs:
= (𝑥𝑥 +
= (𝑥𝑥 +
= (𝑥𝑥 +
2
2)(𝑥𝑥
2
2)(𝑥𝑥
2
2)(𝑥𝑥
)
+ 4)
2𝑥𝑥 + 4)
− 2𝑥𝑥 + 4)
3
3
2
2
𝑎𝑎 − 𝑏𝑏 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Factor:
Write as the
difference of
2 cubes:
Write the binomial
without the cubes:
=
3
8𝑥𝑥
− 27
3
(2𝑥𝑥)
3
−3
= (2𝑥𝑥 − 3)(
)
𝑎𝑎3 + 𝑏𝑏 3 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎2 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 )
= (2𝑥𝑥 − 3)(
Square the first
and last terms:
= (2𝑥𝑥 −
2
3)(4𝑥𝑥
= (2𝑥𝑥 −
2
3)(4𝑥𝑥
Multiply the terms
in the binomial:
Opposite signs:
= (2𝑥𝑥 − 3)(4𝑥𝑥
2
)
The sign of the
last term in the
trinomial is
always positive!
+ 9)
6𝑥𝑥 + 9)
+ 6𝑥𝑥 + 9)
Hint
• Don’t try to factor the trinomial after
factoring the sum or difference of two
cubes.
•=
2𝑥𝑥 − 3 4𝑥𝑥 2 + 6𝑥𝑥 + 9
• If the greatest common factor has already
been taken out, the resulting trinomial
cannot be factored using integers.
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Factor:
Write as a sum of
2 cubes:
Write the binomial
without the cubes:
=
=
6
𝑥𝑥 + 125𝑦𝑦
2
3
(𝑥𝑥 )
2
(𝑥𝑥
3
3
+(5𝑦𝑦)
+ 5𝑦𝑦)(
)
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Square the first
and last terms:
=
=
2
(𝑥𝑥
2
(𝑥𝑥
Multiply the terms
2
in the binomial:= (𝑥𝑥
Opposite signs:
+
+ 5𝑦𝑦)(
4
5𝑦𝑦)(𝑥𝑥
4
2
+
)
2
25𝑦𝑦 )
2
+ 5𝑦𝑦)(𝑥𝑥 5𝑥𝑥 𝑦𝑦 + 25𝑦𝑦 )
= (𝑥𝑥 2 + 5𝑦𝑦)(𝑥𝑥 4 − 5𝑥𝑥 2 𝑦𝑦 + 25𝑦𝑦 2 )
3
3
2
2
𝑎𝑎 − 𝑏𝑏 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Factor:
Write as the
difference of
2 cubes:
Write the binomial
without the cubes:
=
1−
3
(1)
= (1 −
9
3
216𝑥𝑥 𝑦𝑦
−
3
3
6𝑥𝑥 𝑦𝑦
3
6𝑥𝑥 𝑦𝑦)(
)
3
3
2
2
𝑎𝑎 + 𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 )
Square the first
and last terms:
3
= (1 − 6𝑥𝑥 𝑦𝑦)(
3
= (1 − 6𝑥𝑥 𝑦𝑦)(1
Multiply the terms
in the binomial:
3
3
+
)
6
2
36𝑥𝑥 𝑦𝑦 )
6 2
= (1 − 6𝑥𝑥 𝑦𝑦)(1 6𝑥𝑥 𝑦𝑦 + 36𝑥𝑥 𝑦𝑦 )
Opposite signs:
= (1 − 6𝑥𝑥 3 𝑦𝑦)(1 + 6𝑥𝑥 3 𝑦𝑦 + 36𝑥𝑥 6 𝑦𝑦 2 )