Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

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

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x - b$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

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

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $x + b$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Factor out the greatest common factor from each group.

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Step 4.1.1

Group the first two terms and the last two terms.

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

Step 4.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.2

Factor the polynomial by factoring out the greatest common factor, $x - b$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor out the greatest common factor from each group.

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Step 5.1.1

Group the first two terms and the last two terms.

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

Step 5.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.2

Factor the polynomial by factoring out the greatest common factor, $x + b$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x + a$ .

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Step 6.1.1

Factor $x + a$ out of $(x - b) (x + a)$ .

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

Step 6.1.2

Factor $x + a$ out of $(x + b) (x + a)$ .

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

Step 6.1.3

Cancel the common factor.

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

Step 6.1.4

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x - a$ .

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Step 6.2.1

Factor $x - a$ out of $(x + b) (x - a)$ .

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

$\frac{x - b}{x + b} \cdot \frac{x - b}{x + b}$

Step 6.3

Multiply $\frac{x - b}{x + b}$ by $\frac{x - b}{x + b}$ .

$\frac{(x - b) (x - b)}{(x + b) (x + b)}$

Step 7

Raise $x - b$ to the power of $1$ .

$\frac{{(x - b)}^{1} (x - b)}{(x + b) (x + b)}$

Step 8

Raise $x - b$ to the power of $1$ .

$\frac{{(x - b)}^{1} {(x - b)}^{1}}{(x + b) (x + b)}$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - b)}^{1 + 1}}{(x + b) (x + b)}$

Step 10

Add $1$ and $1$ .

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

Step 11

Raise $x + b$ to the power of $1$ .

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

Step 12

Raise $x + b$ to the power of $1$ .

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

Step 13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 14

Add $1$ and $1$ .

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