# Difference of cubes

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Definition.

**Difference of cubes**of two expressions can be found using the following formula:

a^{3} - b^{3} = (a - b)·(a^{2} + ab + b^{2})

## Derivation of the formula of difference of cubes

The proof of the formula is very simple. To prove the formula is sufficient to multiply the expression:

(a - b)·(a

= a

^{2}+ ab + b^{2}) == a

^{3}+ a^{2}b + ab^{2}- ba^{2}- ab^{2}- b^{3}= a^{3}- b^{3}## Applying of difference of cubes formula

Difference of cubes formula convenient to use:

- to factorised
- to simplify expressions

## Examples of task

Example 1.

Factorised x^{3}- 27.

**Solution:** Apply the **difference of cubes formula**.

x

^{3}- 27 = x^{3}- 3^{3}= (x - 3)·(x^{2}+ 3x + 9)Example 2.

Factorised 8x^{3}- 27y

^{6}.

**Solution:** Apply the **difference of cubes formula**.

8x

= (2x - 3y

^{3}- 27y^{6}= (2x)^{3}- (3y^{2})^{3}== (2x - 3y

^{2})·(4x^{2}+ 6xy^{2}+ 9y^{4})Example 3.

Simplify the expression ^{3}- 1

**Solution:** Apply the **difference of cubes formula** in numerator.

^{3}- 1

^{2}+ 3x +1)

^{2}+ 3x +1

Factoring: Some special cases
Square of the sum
Square of the difference
Difference of squares
Cube of sum
Cube of difference
Sum of cubes
Difference of cubes

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