Algebra Examples

$\frac{8 b^{2} - 8 a^{2}}{a^{2} - 2 a b + b^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $8$ out of $8 b^{2} - 8 a^{2}$ .

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Step 1.1.1

Factor $8$ out of $8 b^{2}$ .

$\frac{8 (b^{2}) - 8 a^{2}}{a^{2} - 2 a b + b^{2}}$

Step 1.1.2

Factor $8$ out of $- 8 a^{2}$ .

$\frac{8 (b^{2}) + 8 (- a^{2})}{a^{2} - 2 a b + b^{2}}$

Step 1.1.3

Factor $8$ out of $8 (b^{2}) + 8 (- a^{2})$ .

$\frac{8 (b^{2} - a^{2})}{a^{2} - 2 a b + b^{2}}$

Step 1.2

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

$\frac{8 (b + a) (b - a)}{a^{2} - 2 a b + b^{2}}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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 2.2

Rewrite the polynomial.

$\frac{8 (b + a) (b - a)}{a^{2} - 2 \cdot a \cdot b + b^{2}}$

Step 2.3

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

$\frac{8 (b + a) (b - a)}{{(a - b)}^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

Factor $- 1$ out of $b$ .

$\frac{8 (b + a) (- 1 (- b) - a)}{{(a - b)}^{2}}$

Step 3.1.2

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

$\frac{8 (b + a) (- 1 (- b) - (a))}{{(a - b)}^{2}}$

Step 3.1.3

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

$\frac{8 (b + a) (- 1 (- b + a))}{{(a - b)}^{2}}$

Step 3.1.4

Reorder terms.

$\frac{8 (b + a) (- 1 (a - b))}{{(a - b)}^{2}}$

Step 3.1.5

Factor $a - b$ out of $8 (b + a) (- 1 (a - b))$ .

$\frac{(a - b) (8 (b + a) \cdot (- 1))}{{(a - b)}^{2}}$

Step 3.1.6

Cancel the common factors.

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Step 3.1.6.1

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

$\frac{(a - b) (8 (b + a) \cdot (- 1))}{(a - b) (a - b)}$

Step 3.1.6.2

Cancel the common factor.

$\frac{(a - b) (8 (b + a) \cdot (- 1))}{(a - b) (a - b)}$

Step 3.1.6.3

Rewrite the expression.

$\frac{8 (b + a) \cdot (- 1)}{a - b}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $- 1$ by $8$ .

$\frac{- 8 (b + a)}{a - b}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{8 (b + a)}{a - b}$