# Cube of difference

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Definition.

The **сube of difference**of two expressions is equal to the сube of the first, minus three times the product of the square of the first and the second, plus three times the product of the first and the square of second, minus the сube of the second:

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

## Derivation of the formula of cube of difference

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

(a - b)

= (a - b)·(a

= a

= a

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

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

^{3}- 2a^{2}b + ab^{2}- ba^{2}+ 2b^{2}a - b^{3}== a

^{3}- 3a^{2}b + 3ab^{2}- b^{3}## Applying of cube of difference formula

Cube of difference formula convenient to use:

- to disclose the brackets
- to simplify expressions

## Examples of task

Example 1.

Expand brackets (x - 3)^{3}.

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

(x - 3)

= x

^{3}= x^{3}- 3·3·x^{2}+ 3·3^{2}·x - 3^{3}== x

^{3}- 9x^{2}+ 27x - 27Example 2.

Expand brackets (2x - 3y^{2})

^{3}.

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

(2x - 3y

= (2x)

= 8x

^{2})^{3}== (2x)

^{3}- 3·(2x)^{2}·(3y^{2}) + 3·(2x)·(3y^{2})^{2}- (3y^{2})^{3}== 8x

^{3}- 36x^{2}y^{2}+ 54xy^{4}- 27y^{6}Example 3.

Simplify the expression ^{3}- 27x

^{2}+ 9x - 1

^{2}- 6x + 1

**Solution:** Apply the **cube of difference formula** in numerator and **square of the difference formula** in denominator.

^{3}- 27x

^{2}+ 9x - 1

^{2}- 6x + 1

^{3}

^{2}

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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