Algebra Examples

$\frac{4 x^{3} + 56 x^{2} + 160 x}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{x^{2} - 5 x - 36}$

Step 1

Factor $x^{2} - 5 x - 36$ using the AC method.

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Step 1.1

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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 1.2

Write the factored form using these integers.

$\frac{4 x^{3} + 56 x^{2} + 160 x}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $4 x$ out of $4 x^{3} + 56 x^{2} + 160 x$ .

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Step 2.1.1

Factor $4 x$ out of $4 x^{3}$ .

$\frac{4 x (x^{2}) + 56 x^{2} + 160 x}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2.1.2

Factor $4 x$ out of $56 x^{2}$ .

$\frac{4 x (x^{2}) + 4 x (14 x) + 160 x}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2.1.3

Factor $4 x$ out of $160 x$ .

$\frac{4 x (x^{2}) + 4 x (14 x) + 4 x (40)}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2.1.4

Factor $4 x$ out of $4 x (x^{2}) + 4 x (14 x)$ .

$\frac{4 x (x^{2} + 14 x) + 4 x (40)}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2.1.5

Factor $4 x$ out of $4 x (x^{2} + 14 x) + 4 x (40)$ .

$\frac{4 x (x^{2} + 14 x + 40)}{x + 10} \cdot \frac{81 - x^{2}}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 2.2

Factor $x^{2} + 14 x + 40$ 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 $40$ and whose sum is $14$ .

$4, 10$

Step 2.2.2

Write the factored form using these integers.

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

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 $a = 9$ and $b = x$ .

$\frac{4 x (x + 4) (x + 10)}{x + 10} \cdot \frac{(9 + x) (9 - x)}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 10$ .

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Step 4.1.1

Cancel the common factor.

$\frac{4 x (x + 4) (x + 10)}{x + 10} \cdot \frac{(9 + x) (9 - x)}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 4.1.2

Divide $4 x (x + 4)$ by $1$ .

$4 x (x + 4) \cdot \frac{(9 + x) (9 - x)}{- x - 9} \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 4.2

Cancel the common factor of $x + 4$ .

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Step 4.2.1

Factor $x + 4$ out of $4 x (x + 4) \cdot \frac{(9 + x) (9 - x)}{- x - 9}$ .

$(x + 4) (4 x \cdot \frac{(9 + x) (9 - x)}{- x - 9}) \cdot \frac{3 x}{(x - 9) (x + 4)}$

Step 4.2.2

Factor $x + 4$ out of $(x - 9) (x + 4)$ .

$(x + 4) (4 x \cdot \frac{(9 + x) (9 - x)}{- x - 9}) \cdot \frac{3 x}{(x + 4) (x - 9)}$

Step 4.2.3

Cancel the common factor.

$(x + 4) (4 x \cdot \frac{(9 + x) (9 - x)}{- x - 9}) \cdot \frac{3 x}{(x + 4) (x - 9)}$

Step 4.2.4

Rewrite the expression.

$4 x \cdot \frac{(9 + x) (9 - x)}{- x - 9} \cdot \frac{3 x}{x - 9}$

Step 4.3

Combine $4$ and $\frac{(9 + x) (9 - x)}{- x - 9}$ .

$x \cdot \frac{4 ((9 + x) (9 - x))}{- x - 9} \cdot \frac{3 x}{x - 9}$

Step 4.4

Combine $x$ and $\frac{4 ((9 + x) (9 - x))}{- x - 9}$ .

$\frac{x (4 ((9 + x) (9 - x)))}{- x - 9} \cdot \frac{3 x}{x - 9}$

Step 4.5

Multiply $\frac{x (4 ((9 + x) (9 - x)))}{- x - 9}$ by $\frac{3 x}{x - 9}$ .

$\frac{x (4 ((9 + x) (9 - x))) (3 x)}{(- x - 9) (x - 9)}$

Step 5

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

$\frac{x^{1} x (4 ((9 + x) (9 - x))) \cdot 3}{(- x - 9) (x - 9)}$

Step 6

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

$\frac{x^{1} x^{1} (4 ((9 + x) (9 - x))) \cdot 3}{(- x - 9) (x - 9)}$

Step 7

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

$\frac{x^{1 + 1} (4 ((9 + x) (9 - x))) \cdot 3}{(- x - 9) (x - 9)}$

Step 8

Simplify the expression.

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Step 8.1

Add $1$ and $1$ .

$\frac{x^{2} (4 ((9 + x) (9 - x))) \cdot 3}{(- x - 9) (x - 9)}$

Step 8.2

Multiply $3$ by $4$ .

$\frac{x^{2} (12 ((9 + x) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9

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

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Step 9.1

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

$\frac{x^{2} (12 ((- 1 (- 9) + x) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9.2

Factor $- 1$ out of $x$ .

$\frac{x^{2} (12 ((- 1 (- 9) - 1 (- x)) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9.3

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

$\frac{x^{2} (12 (- 1 (- 9 - x) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9.4

Reorder terms.

$\frac{x^{2} (12 (- 1 (- x - 9) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9.5

Cancel the common factor.

$\frac{x^{2} (12 (- 1 (- x - 9) (9 - x)))}{(- x - 9) (x - 9)}$

Step 9.6

Rewrite the expression.

$\frac{x^{2} (12 (- 1 (9 - x)))}{x - 9}$

Step 10

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

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Step 10.1

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

$\frac{x^{2} (12 (- 1 (- 1 (- 9) - x)))}{x - 9}$

Step 10.2

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

$\frac{x^{2} (12 (- 1 (- 1 (- 9) - (x))))}{x - 9}$

Step 10.3

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

$\frac{x^{2} (12 (- 1 (- 1 (- 9 + x))))}{x - 9}$

Step 10.4

Reorder terms.

$\frac{x^{2} (12 (- 1 (- 1 (x - 9))))}{x - 9}$

Step 10.5

Cancel the common factor.

$\frac{x^{2} (12 (- 1 (- 1 (x - 9))))}{x - 9}$

Step 10.6

Divide $x^{2} (12 (- 1 \cdot (- 1)))$ by $1$ .

$x^{2} (12 (- 1 \cdot (- 1)))$

Step 11

Multiply $12 (- 1 \cdot (- 1))$ .

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Step 11.1

Multiply $- 1$ by $- 1$ .

$x^{2} (12 \cdot 1)$

Step 11.2

Multiply $12$ by $1$ .

$x^{2} \cdot 12$

Step 12

Move $12$ to the left of $x^{2}$ .

$12 x^{2}$