In Algebra 1, you worked with factoring the differenceof two perfect squares.
a2 - b2 = (a - b)(a + b)
note The sum of two perfect squares, a2 + b2, does not factor.

In Algebra 2, we will extend our factoring skills
to factoring both the differenceand the sumof two perfect cubes.

Factor the Sum of Perfect Cubes:
a3 + b3 = (a + b)(a2 - ab + b2)
Factor the Difference of Perfect Cubes:
a3 - b3 = (a - b)(a2 + ab + b2)

hintgal
When trying to remember these patterns, remember that the first binomial term in the factored form in each pattern keeps the same sign as the sign between the perfect cubes. The sign separating the first and second terms of the trinomial is opposite the sign between the perfect cubes, and the last sign in the trinomial is always positive.
(Note that each formula has only one negative sign in the solution.)

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If you remember that the first part of the factoring (the binomial part) is the sum (or difference) of a and b,
you can always find the rest of the factored answer by using long division.

Let's see that these formulas are actually true.

Check a3 + b3 formula:
show1
 
Check a3 - b3 formula:
show2

When working with perfect cubes look for:
perfect cube numerical values: 8, 27, 64, 125, 216, 343, 512, ...
powers for perfect cube algebraic values: cubepowers

pin1
Factor: 8x3 - 27
Is this expression the difference of perfect cubes?
Look for perfect cubes and appropriate powers.
8x3 and 27 are perfect cubes: 8x3 = (2x)3 and 27 = 33
So, yes, this is the difference of perfect cubes.
(2x)3 - 33
a = 2x, b = 3
Use formula: a3 - b3 = (a - b)(a2 + ab + b2).
Be sure to use parentheses to avoid problems.
(2x - 3)((2x)2 + (2x)(3) + 32)
Remove the inner parentheses on the trinomial expression. This is your factored answer.
(2x - 3)(4x2 + 6x + 9)

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pin2
Factor: 64x9 + 125
64x9 = (4x3)3 and 125 = 53
This expression is the sum of perfect cubes.
(4x3)3 + 53
a = 4x3 and b = 5
formula: a3 + b3 = (a + b)(a2 - ab + b2)
(4x3 + 5)((4x3)2 - (4x3)(5)+ 52)
Simplify the trinomial to get the factored form.
(4x3 + 5)(16x6- 20x3+ 25)

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pin3
Factor: 56m3 - 7n3
At first glance, this may not look like the difference of perfect cubes. BUT, if we factor out the common factor of 7, we will find a hidden difference of perfect cubes.
7(8m3 - n3)
a = 2m and b = n
Use formula: a3 - b3 = (a - b)(a2 + ab + b2).
7(2m - n)((4m)2 + 2mn + n2)
Solution:
7(2m - n)(4m2 + 2mn + n2)

 

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