# Sum of cubes

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Definition.

**Sum 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 sum 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 sum of cubes formula

Sum of cubes formula convenient to use:

- to factorised
- to simplify expressions

## Examples of task

Example 1.

Factorised x^{3}+ 27.

**Solution:** Apply the **sum 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 **sum 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 **sum 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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