Algebra Examples

$\frac{x^{3} + 10 x^{2} + 25 x}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \div \frac{x^{2} - x - 56}{x + 7}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3} + 10 x^{2} + 25 x}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $x$ out of $x^{3} + 10 x^{2} + 25 x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} + 10 x^{2} + 25 x}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.1.2

Factor $x$ out of $10 x^{2}$ .

$\frac{x \cdot x^{2} + x (10 x) + 25 x}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.1.3

Factor $x$ out of $25 x$ .

$\frac{x \cdot x^{2} + x (10 x) + x \cdot 25}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.1.4

Factor $x$ out of $x \cdot x^{2} + x (10 x)$ .

$\frac{x (x^{2} + 10 x) + x \cdot 25}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.1.5

Factor $x$ out of $x (x^{2} + 10 x) + x \cdot 25$ .

$\frac{x (x^{2} + 10 x + 25)}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{x (x^{2} + 10 x + 5^{2})}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 2.2.3

Rewrite the polynomial.

$\frac{x (x^{2} + 2 \cdot x \cdot 5 + 5^{2})}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{x {(x + 5)}^{2}}{x^{2} + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 3

Factor $x$ out of $x^{2} + 5 x$ .

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Step 3.1

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

$\frac{x {(x + 5)}^{2}}{x \cdot x + 5 x} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 3.2

Factor $x$ out of $5 x$ .

$\frac{x {(x + 5)}^{2}}{x \cdot x + x \cdot 5} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 3.3

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

$\frac{x {(x + 5)}^{2}}{x (x + 5)} \cdot \frac{x^{2} - 64}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $64$ as $8^{2}$ .

$\frac{x {(x + 5)}^{2}}{x (x + 5)} \cdot \frac{x^{2} - 8^{2}}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 4.2

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 $b = 8$ .

$\frac{x {(x + 5)}^{2}}{x (x + 5)} \cdot \frac{(x + 8) (x - 8)}{x^{2} + 13 x + 40} \frac{x + 7}{x^{2} - x - 56}$

Step 5

Factor $x^{2} + 13 x + 40$ using the AC method.

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Step 5.1

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 $40$ and whose sum is $13$ .

$5, 8$

Step 5.2

Write the factored form using these integers.

$\frac{x {(x + 5)}^{2}}{x (x + 5)} \cdot \frac{(x + 8) (x - 8)}{(x + 5) (x + 8)} \frac{x + 7}{x^{2} - x - 56}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{x {(x + 5)}^{2} ((x + 8) (x - 8))}{x (x + 5) ((x + 5) (x + 8))} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.2

Cancel the common factor of $x$ .

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Step 6.2.1

Cancel the common factor.

$\frac{x {(x + 5)}^{2} ((x + 8) (x - 8))}{x (x + 5) ((x + 5) (x + 8))} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.2.2

Rewrite the expression.

$\frac{{(x + 5)}^{2} ((x + 8) (x - 8))}{(x + 5) ((x + 5) (x + 8))} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.3

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 6.3.1

Factor $x + 5$ out of ${(x + 5)}^{2} ((x + 8) (x - 8))$ .

$\frac{(x + 5) ((x + 5) ((x + 8) (x - 8)))}{(x + 5) ((x + 5) (x + 8))} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Cancel the common factor.

$\frac{(x + 5) ((x + 5) ((x + 8) (x - 8)))}{(x + 5) ((x + 5) (x + 8))} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.3.2.2

Rewrite the expression.

$\frac{(x + 5) ((x + 8) (x - 8))}{(x + 5) (x + 8)} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.4

Cancel the common factor of $x + 5$ .

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Step 6.4.1

Cancel the common factor.

$\frac{(x + 5) ((x + 8) (x - 8))}{(x + 5) (x + 8)} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.4.2

Rewrite the expression.

$\frac{(x + 8) (x - 8)}{x + 8} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.5

Cancel the common factor of $x + 8$ .

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Step 6.5.1

Cancel the common factor.

$\frac{(x + 8) (x - 8)}{x + 8} \cdot \frac{x + 7}{x^{2} - x - 56}$

Step 6.5.2

Divide $x - 8$ by $1$ .

$(x - 8) \frac{x + 7}{x^{2} - x - 56}$

Step 7

Factor $x^{2} - x - 56$ using the AC method.

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Step 7.1

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 $- 56$ and whose sum is $- 1$ .

$- 8, 7$

Step 7.2

Write the factored form using these integers.

$(x - 8) \frac{x + 7}{(x - 8) (x + 7)}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $x - 8$ .

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Step 8.1.1

Cancel the common factor.

$(x - 8) \frac{x + 7}{(x - 8) (x + 7)}$

Step 8.1.2

Rewrite the expression.

$\frac{x + 7}{x + 7}$

Step 8.2

Cancel the common factor of $x + 7$ .

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Step 8.2.1

Cancel the common factor.

$\frac{x + 7}{x + 7}$

Step 8.2.2

Rewrite the expression.

$1$