Algebra Examples

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x^{3} + 27 x^{2} - 630 x}{- x - 10}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $9 x$ out of $9 x^{3} + 27 x^{2} - 630 x$ .

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Step 1.1.1

Factor $9 x$ out of $9 x^{3}$ .

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x^{2}) + 27 x^{2} - 630 x}{- x - 10}$

Step 1.1.2

Factor $9 x$ out of $27 x^{2}$ .

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x^{2}) + 9 x (3 x) - 630 x}{- x - 10}$

Step 1.1.3

Factor $9 x$ out of $- 630 x$ .

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x^{2}) + 9 x (3 x) + 9 x (- 70)}{- x - 10}$

Step 1.1.4

Factor $9 x$ out of $9 x (x^{2}) + 9 x (3 x)$ .

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x^{2} + 3 x) + 9 x (- 70)}{- x - 10}$

Step 1.1.5

Factor $9 x$ out of $9 x (x^{2} + 3 x) + 9 x (- 70)$ .

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x^{2} + 3 x - 70)}{- x - 10}$

Step 1.2

Factor $x^{2} + 3 x - 70$ using the AC method.

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Step 1.2.1

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 $- 70$ and whose sum is $3$ .

$- 7, 10$

Step 1.2.2

Write the factored form using these integers.

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x ((x - 7) (x + 10))}{- x - 10}$

$\frac{81 - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$\frac{9^{2} - x^{2}}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = x$ .

$\frac{(9 + x) (9 - x)}{x^{2} - 81} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $81$ as $9^{2}$ .

$\frac{(9 + x) (9 - x)}{x^{2} - 9^{2}} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 3.2

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 $b = 9$ .

$\frac{(9 + x) (9 - x)}{(x + 9) (x - 9)} \cdot \frac{4}{x^{2} - 16 x + 63} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 4

Factor $x^{2} - 16 x + 63$ using the AC method.

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Step 4.1

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 $63$ and whose sum is $- 16$ .

$- 9, - 7$

Step 4.2

Write the factored form using these integers.

$\frac{(9 + x) (9 - x)}{(x + 9) (x - 9)} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $9 + x$ and $x + 9$ .

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Step 5.1.1

Reorder terms.

$\frac{(x + 9) (9 - x)}{(x + 9) (x - 9)} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.1.2

Cancel the common factor.

$\frac{(x + 9) (9 - x)}{(x + 9) (x - 9)} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.1.3

Rewrite the expression.

$\frac{9 - x}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2

Cancel the common factor of $9 - x$ and $x - 9$ .

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Step 5.2.1

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- 1 (- 9) - x}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 9) - (x)}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2.3

Factor $- 1$ out of $- 1 (- 9) - (x)$ .

$\frac{- 1 (- 9 + x)}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2.4

Reorder terms.

$\frac{- 1 (x - 9)}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2.5

Cancel the common factor.

$\frac{- 1 (x - 9)}{x - 9} \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.2.6

Divide $- 1$ by $1$ .

$- 1 \cdot \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.3

Rewrite $- 1 \frac{4}{(x - 9) (x - 7)}$ as $- \frac{4}{(x - 9) (x - 7)}$ .

$- \frac{4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.4

Cancel the common factor of $x - 7$ .

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Step 5.4.1

Move the leading negative in $- \frac{4}{(x - 9) (x - 7)}$ into the numerator.

$\frac{- 4}{(x - 9) (x - 7)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.4.2

Factor $x - 7$ out of $(x - 9) (x - 7)$ .

$\frac{- 4}{(x - 7) (x - 9)} \cdot \frac{9 x (x - 7) (x + 10)}{- x - 10}$

Step 5.4.3

Factor $x - 7$ out of $9 x (x - 7) (x + 10)$ .

$\frac{- 4}{(x - 7) (x - 9)} \cdot \frac{(x - 7) (9 x (x + 10))}{- x - 10}$

Step 5.4.4

Cancel the common factor.

$\frac{- 4}{(x - 7) (x - 9)} \cdot \frac{(x - 7) (9 x (x + 10))}{- x - 10}$

Step 5.4.5

Rewrite the expression.

$\frac{- 4}{x - 9} \cdot \frac{9 x (x + 10)}{- x - 10}$

Step 5.5

Multiply $\frac{- 4}{x - 9}$ by $\frac{9 x (x + 10)}{- x - 10}$ .

$\frac{- 4 (9 x (x + 10))}{(x - 9) (- x - 10)}$

Step 5.6

Multiply $9$ by $- 4$ .

$\frac{- 36 (x (x + 10))}{(x - 9) (- x - 10)}$

Step 5.7

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

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Step 5.7.1

Factor $- 1$ out of $x$ .

$\frac{- 36 (x (- 1 (- x) + 10))}{(x - 9) (- x - 10)}$

Step 5.7.2

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

$\frac{- 36 (x (- 1 (- x) - 1 (- 10)))}{(x - 9) (- x - 10)}$

Step 5.7.3

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

$\frac{- 36 (x (- 1 (- x - 10)))}{(x - 9) (- x - 10)}$

Step 5.7.4

Cancel the common factor.

$\frac{- 36 x (- 1 (- x - 10))}{(x - 9) (- x - 10)}$

Step 5.7.5

Rewrite the expression.

$\frac{- 36 x \cdot (- 1)}{x - 9}$

Step 5.8

Multiply $- 1$ by $- 36$ .

$\frac{36 x}{x - 9}$