4. The Sum and Difference of Cubes
We came across these expressions earlier (in the section Special Products involving Cubes):
x3 + y3 = (x + y)(x2 − xy + y2) [Sum of two cubes]
x3 − y3 = (x − y)(x2 + xy + y2) [Difference of 2 cubes]
Where do these come from? If you multiply out the right side of each, you'll get the left side of the equation.
Note: We cannot factor the right hand sides any further.
We use the above formulas to factor expressions involving cubes, as in the following example.
Factor `64x^3 + 125`
We use the Sum of 2 Cubes formula given above.
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)[(4x)2 − (4x)(5) + (5)2]
= (4x + 5)(16x2 − 20x + 25)
As mentioned above, we cannot factor the expression in the second bracket any further. It looks like it could be factored to give `(4x-5)^2`, however, when we expand this it gives:
This "perfect square trinomial" is not the same as the expression we obtained when factoring the sum of 2 cubes.
(1) x3 + 27
(2) 3m3 − 81
Didn't find what you are looking for on this page? Try search:
Online Algebra Solver
This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)
Go to: Online algebra solver
Ready for a break?
Play a math game.
(Well, not really a math game, but each game was made using math...)
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Algebra Lessons on DVD
Easy to understand algebra lessons on DVD. See samples before you commit.
More info: Algebra videos