Algebra Examples

$\frac{a^{3} - b^{3}}{{(a - b)}^{2}} + \frac{3 a b}{b - a}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = a$ and $b = b$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{{(a - b)}^{2}} + \frac{3 a b}{b - a}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $a - b$ out of ${(a - b)}^{2}$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{(a - b) (a - b)} + \frac{3 a b}{b - a}$

Step 1.1.2.2

Cancel the common factor.

$\frac{(a - b) (a^{2} + a b + b^{2})}{(a - b) (a - b)} + \frac{3 a b}{b - a}$

Step 1.1.2.3

Rewrite the expression.

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{3 a b}{b - a}$

Step 1.2

Simplify terms.

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Step 1.2.1

Factor $- 1$ out of $b$ .

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{3 a b}{- 1 (- b) - a}$

Step 1.2.2

Factor $- 1$ out of $- a$ .

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{3 a b}{- 1 (- b) - (a)}$

Step 1.2.3

Factor $- 1$ out of $- 1 (- b) - (a)$ .

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{3 a b}{- 1 (- b + a)}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Move a negative from the denominator of $\frac{3 a b}{- 1 (- b + a)}$ to the numerator.

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{- (3 a b)}{- b + a}$

Step 1.2.4.2

Reorder terms.

$\frac{a^{2} + a b + b^{2}}{a - b} + \frac{- (3 a b)}{a - b}$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{a^{2} + a b + b^{2} - (3 a b)}{a - b}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $3$ .

$\frac{a^{2} + a b + b^{2} - 3 a b}{a - b}$

Step 2.2

Subtract $3 a b$ from $a b$ .

$\frac{a^{2} + b^{2} - 2 a b}{a - b}$

Step 2.3

Factor using the perfect square rule.

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Step 2.3.1

Rearrange terms.

$\frac{a^{2} - 2 a b + b^{2}}{a - b}$

Step 2.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot a \cdot b$

Step 2.3.3

Rewrite the polynomial.

$\frac{a^{2} - 2 \cdot a \cdot b + b^{2}}{a - b}$

Step 2.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = b$ .

$\frac{{(a - b)}^{2}}{a - b}$

Step 3

Cancel the common factor of ${(a - b)}^{2}$ and $a - b$ .

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Step 3.1

Factor $a - b$ out of ${(a - b)}^{2}$ .

$\frac{(a - b) (a - b)}{a - b}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\frac{(a - b) (a - b)}{(a - b) \cdot 1}$

Step 3.2.2

Cancel the common factor.

$\frac{(a - b) (a - b)}{(a - b) \cdot 1}$

Step 3.2.3

Rewrite the expression.

$\frac{a - b}{1}$

Step 3.2.4

Divide $a - b$ by $1$ .

$a - b$