Algebra Examples

$\frac{2 x^{3} + 20 x^{2} + 32 x}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \div \frac{- x - 10}{x^{2} + x - 90}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x^{3} + 20 x^{2} + 32 x}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2 x$ out of $2 x^{3} + 20 x^{2} + 32 x$ .

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Step 2.1.1

Factor $2 x$ out of $2 x^{3}$ .

$\frac{2 x (x^{2}) + 20 x^{2} + 32 x}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2.1.2

Factor $2 x$ out of $20 x^{2}$ .

$\frac{2 x (x^{2}) + 2 x (10 x) + 32 x}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2.1.3

Factor $2 x$ out of $32 x$ .

$\frac{2 x (x^{2}) + 2 x (10 x) + 2 x (16)}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2.1.4

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

$\frac{2 x (x^{2} + 10 x) + 2 x (16)}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2.1.5

Factor $2 x$ out of $2 x (x^{2} + 10 x) + 2 x (16)$ .

$\frac{2 x (x^{2} + 10 x + 16)}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 2.2

Factor $x^{2} + 10 x + 16$ using the AC method.

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Step 2.2.1

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 $16$ and whose sum is $10$ .

$2, 8$

Step 2.2.2

Write the factored form using these integers.

$\frac{2 x ((x + 2) (x + 8))}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

$\frac{2 x (x + 2) (x + 8)}{x^{2} - 7 x - 18} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 3

Factor $x^{2} - 7 x - 18$ 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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 3.2

Write the factored form using these integers.

$\frac{2 x (x + 2) (x + 8)}{(x - 9) (x + 2)} \cdot \frac{- 6}{6 x + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 4

Simplify terms.

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Step 4.1

Factor $6$ out of $6 x + 48$ .

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Step 4.1.1

Factor $6$ out of $6 x$ .

$\frac{2 x (x + 2) (x + 8)}{(x - 9) (x + 2)} \cdot \frac{- 6}{6 (x) + 48} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.1.2

Factor $6$ out of $48$ .

$\frac{2 x (x + 2) (x + 8)}{(x - 9) (x + 2)} \cdot \frac{- 6}{6 x + 6 \cdot 8} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.1.3

Factor $6$ out of $6 x + 6 \cdot 8$ .

$\frac{2 x (x + 2) (x + 8)}{(x - 9) (x + 2)} \cdot \frac{- 6}{6 (x + 8)} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.2

Cancel the common factor of $2 (x + 8)$ .

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Step 4.2.1

Factor $2 (x + 8)$ out of $2 x (x + 2) (x + 8)$ .

$\frac{2 (x + 8) (x (x + 2))}{(x - 9) (x + 2)} \cdot \frac{- 6}{6 (x + 8)} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.2.2

Factor $2 (x + 8)$ out of $6 (x + 8)$ .

$\frac{2 (x + 8) (x (x + 2))}{(x - 9) (x + 2)} \cdot \frac{- 6}{2 (x + 8) (3)} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.2.3

Cancel the common factor.

$\frac{2 (x + 8) (x (x + 2))}{(x - 9) (x + 2)} \cdot \frac{- 6}{2 (x + 8) \cdot 3} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.2.4

Rewrite the expression.

$\frac{x (x + 2)}{(x - 9) (x + 2)} \cdot \frac{- 6}{3} \frac{x^{2} + x - 90}{- x - 10}$

Step 4.3

Multiply $\frac{x (x + 2)}{(x - 9) (x + 2)}$ by $\frac{- 6}{3}$ .

$\frac{x (x + 2) \cdot - 6}{(x - 9) (x + 2) \cdot 3} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.4

Cancel the common factor of $x + 2$ .

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Step 4.4.1

Cancel the common factor.

$\frac{x (x + 2) \cdot - 6}{(x - 9) (x + 2) \cdot 3} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.4.2

Rewrite the expression.

$\frac{x \cdot - 6}{(x - 9) \cdot 3} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.5

Cancel the common factor of $- 6$ and $3$ .

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Step 4.5.1

Factor $3$ out of $x \cdot - 6$ .

$\frac{3 (x \cdot - 2)}{(x - 9) \cdot 3} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $3$ out of $(x - 9) \cdot 3$ .

$\frac{3 (x \cdot - 2)}{3 \cdot (x - 9)} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.5.2.2

Cancel the common factor.

$\frac{3 (x \cdot - 2)}{3 \cdot (x - 9)} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.5.2.3

Rewrite the expression.

$\frac{x \cdot - 2}{x - 9} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.6

Simplify the expression.

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Step 4.6.1

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

$\frac{- 2 \cdot x}{x - 9} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 4.6.2

Move the negative in front of the fraction.

$- \frac{2 \cdot x}{x - 9} \cdot \frac{x^{2} + x - 90}{- x - 10}$

Step 5

Factor $x^{2} + x - 90$ 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 $- 90$ and whose sum is $1$ .

$- 9, 10$

Step 5.2

Write the factored form using these integers.

$- \frac{2 x}{x - 9} \cdot \frac{(x - 9) (x + 10)}{- x - 10}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 9$ .

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Step 6.1.1

Move the leading negative in $- \frac{2 x}{x - 9}$ into the numerator.

$\frac{- 2 x}{x - 9} \cdot \frac{(x - 9) (x + 10)}{- x - 10}$

Step 6.1.2

Cancel the common factor.

$\frac{- 2 x}{x - 9} \cdot \frac{(x - 9) (x + 10)}{- x - 10}$

Step 6.1.3

Rewrite the expression.

$- 2 x \frac{x + 10}{- x - 10}$

Step 6.2

Combine $- 2$ and $\frac{x + 10}{- x - 10}$ .

$x \frac{- 2 (x + 10)}{- x - 10}$

Step 6.3

Combine $x$ and $\frac{- 2 (x + 10)}{- x - 10}$ .

$\frac{x (- 2 (x + 10))}{- x - 10}$

Step 6.4

Cancel the common factor of $x + 10$ and $- x - 10$ .

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Step 6.4.1

Factor $- 1$ out of $x$ .

$\frac{x (- 2 (- 1 (- x) + 10))}{- x - 10}$

Step 6.4.2

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{x (- 2 (- 1 (- x) - 1 (- 10)))}{- x - 10}$

Step 6.4.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 10)$ .

$\frac{x (- 2 (- 1 (- x - 10)))}{- x - 10}$

Step 6.4.4

Cancel the common factor.

$\frac{x (- 2 (- 1 (- x - 10)))}{- x - 10}$

Step 6.4.5

Divide $x (- 2 \cdot (- 1))$ by $1$ .

$x (- 2 \cdot (- 1))$

Step 6.5

Simplify the expression.

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Step 6.5.1

Multiply $- 2$ by $- 1$ .

$x \cdot 2$

Step 6.5.2

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

$2 x$