# 4. The Sum and Difference of Cubes

We came across these expressions earlier (in the section Special Products involving Cubes):

x^{3}+y^{3}= (x+y)(x^{2}−xy+y^{2}) [Sum of two cubes]

x^{3}−y^{3}= (x−y)(x^{2}+xy+y^{2}) [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.

### Example

Factor `64x^3 + 125`

**Answer:**

We use the Sum of 2 Cubes formula given above.

64*x*^{3} + 125

= (4

x)^{3}+ (5)^{3}= (4

x+ 5)[(4x)^{2}− (4x)(5) + (5)^{2}]= (4

x+ 5)(16x^{2}− 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:

`(4x-5)^2=16x^2-40x+25`

This "perfect square trinomial" is not the same as the expression we obtained when factoring the sum of 2 cubes.

### Exercises

Factor:

**(1) ***x*^{3} + 27

**(2) **3*m*^{3} − 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!

### Share IntMath!

### Algebra Lessons on DVD

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos