Algebra Examples

$\frac{x^{3} - 8}{x^{2} + 3 x - 10} \div \frac{8 x - 40}{x^{2} - 25}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3} - 8}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $8$ as $2^{3}$ .

$\frac{x^{3} - 2^{3}}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 2.2

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 $a = x$ and $b = 2$ .

$\frac{(x - 2) (x^{2} + x \cdot 2 + 2^{2})}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 2.3

Simplify.

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Step 2.3.1

Move $2$ to the left of $x$ .

$\frac{(x - 2) (x^{2} + 2 \cdot x + 2^{2})}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 2.3.2

Raise $2$ to the power of $2$ .

$\frac{(x - 2) (x^{2} + 2 x + 4)}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 3

Factor $x^{2} + 3 x - 10$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 2) (x^{2} + 2 x + 4)}{(x - 2) (x + 5)} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 4

Cancel the common factor of $x - 2$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 2) (x^{2} + 2 x + 4)}{(x - 2) (x + 5)} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 4.2

Rewrite the expression.

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{x^{2} - 25}{8 x - 40}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{x^{2} - 5^{2}}{8 x - 40}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{(x + 5) (x - 5)}{8 x - 40}$

Step 6

Simplify terms.

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Step 6.1

Factor $8$ out of $8 x - 40$ .

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Step 6.1.1

Factor $8$ out of $8 x$ .

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{(x + 5) (x - 5)}{8 (x) - 40}$

Step 6.1.2

Factor $8$ out of $- 40$ .

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{(x + 5) (x - 5)}{8 x + 8 \cdot - 5}$

Step 6.1.3

Factor $8$ out of $8 x + 8 \cdot - 5$ .

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{(x + 5) (x - 5)}{8 (x - 5)}$

Step 6.2

Cancel the common factor of $x + 5$ .

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Step 6.2.1

Cancel the common factor.

$\frac{x^{2} + 2 x + 4}{x + 5} \cdot \frac{(x + 5) (x - 5)}{8 (x - 5)}$

Step 6.2.2

Rewrite the expression.

$(x^{2} + 2 x + 4) \frac{x - 5}{8 (x - 5)}$

Step 6.3

Cancel the common factor of $x - 5$ .

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Step 6.3.1

Cancel the common factor.

$(x^{2} + 2 x + 4) \frac{x - 5}{8 (x - 5)}$

Step 6.3.2

Rewrite the expression.

$(x^{2} + 2 x + 4) \frac{1}{8}$

Step 6.4

Apply the distributive property.

$x^{2} \frac{1}{8} + 2 x \frac{1}{8} + 4 (\frac{1}{8})$

Step 7

Simplify.

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Step 7.1

Combine $x^{2}$ and $\frac{1}{8}$ .

$\frac{x^{2}}{8} + 2 x \frac{1}{8} + 4 (\frac{1}{8})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $2 x$ .

$\frac{x^{2}}{8} + 2 (x) \frac{1}{8} + 4 (\frac{1}{8})$

Step 7.2.2

Factor $2$ out of $8$ .

$\frac{x^{2}}{8} + 2 (x) \frac{1}{2 (4)} + 4 (\frac{1}{8})$

Step 7.2.3

Cancel the common factor.

$\frac{x^{2}}{8} + 2 x \frac{1}{2 \cdot 4} + 4 (\frac{1}{8})$

Step 7.2.4

Rewrite the expression.

$\frac{x^{2}}{8} + x \frac{1}{4} + 4 (\frac{1}{8})$

Step 7.3

Combine $x$ and $\frac{1}{4}$ .

$\frac{x^{2}}{8} + \frac{x}{4} + 4 (\frac{1}{8})$

Step 7.4

Cancel the common factor of $4$ .

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Step 7.4.1

Factor $4$ out of $8$ .

$\frac{x^{2}}{8} + \frac{x}{4} + 4 \frac{1}{4 (2)}$

Step 7.4.2

Cancel the common factor.

$\frac{x^{2}}{8} + \frac{x}{4} + 4 \frac{1}{4 \cdot 2}$

Step 7.4.3

Rewrite the expression.

$\frac{x^{2}}{8} + \frac{x}{4} + \frac{1}{2}$

Step 8

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2}}{8} + \frac{x}{4} \cdot \frac{2}{2} + \frac{1}{2}$

Step 9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{x}{4}$ by $\frac{2}{2}$ .

$\frac{x^{2}}{8} + \frac{x \cdot 2}{4 \cdot 2} + \frac{1}{2}$

Step 9.2

Multiply $4$ by $2$ .

$\frac{x^{2}}{8} + \frac{x \cdot 2}{8} + \frac{1}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{x^{2} + x \cdot 2}{8} + \frac{1}{2}$

Step 11

Factor $x$ out of $x^{2} + x \cdot 2$ .

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Step 11.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + x \cdot 2}{8} + \frac{1}{2}$

Step 11.2

Factor $x$ out of $x \cdot 2$ .

$\frac{x \cdot x + x (2)}{8} + \frac{1}{2}$

Step 11.3

Factor $x$ out of $x \cdot x + x (2)$ .

$\frac{x (x + 2)}{8} + \frac{1}{2}$

Step 12

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x (x + 2)}{8} + \frac{1}{2} \cdot \frac{4}{4}$

Step 13

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$\frac{x (x + 2)}{8} + \frac{4}{2 \cdot 4}$

Step 13.2

Multiply $2$ by $4$ .

$\frac{x (x + 2)}{8} + \frac{4}{8}$

Step 14

Combine the numerators over the common denominator.

$\frac{x (x + 2) + 4}{8}$

Step 15

Simplify the numerator.

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Step 15.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 + 4}{8}$

Step 15.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 + 4}{8}$

Step 15.3

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 x + 4}{8}$