Algebra Examples

$\frac{2 x^{5} - 50 x^{3}}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x^{3}$ out of $2 x^{5} - 50 x^{3}$ .

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Step 1.1.1

Factor $2 x^{3}$ out of $2 x^{5}$ .

$\frac{2 x^{3} (x^{2}) - 50 x^{3}}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 1.1.2

Factor $2 x^{3}$ out of $- 50 x^{3}$ .

$\frac{2 x^{3} (x^{2}) + 2 x^{3} (- 25)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 1.1.3

Factor $2 x^{3}$ out of $2 x^{3} (x^{2}) + 2 x^{3} (- 25)$ .

$\frac{2 x^{3} (x^{2} - 25)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 1.2

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

$\frac{2 x^{3} (x^{2} - 5^{2})}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 1.3

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 $b = 5$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 2

Factor $x^{2} + 4 x - 45$ using the AC method.

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Step 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 $- 45$ and whose sum is $4$ .

$- 5, 9$

Step 2.2

Write the factored form using these integers.

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{x^{2} - 9^{2}}{6 x^{2} - 24 x - 270}$

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 $b = 9$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 x^{2} - 24 x - 270}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $6$ out of $6 x^{2} - 24 x - 270$ .

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Step 4.1.1

Factor $6$ out of $6 x^{2}$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2}) - 24 x - 270}$

Step 4.1.2

Factor $6$ out of $- 24 x$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2}) + 6 (- 4 x) - 270}$

Step 4.1.3

Factor $6$ out of $- 270$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 x^{2} + 6 (- 4 x) + 6 \cdot - 45}$

Step 4.1.4

Factor $6$ out of $6 x^{2} + 6 (- 4 x)$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2} - 4 x) + 6 \cdot - 45}$

Step 4.1.5

Factor $6$ out of $6 (x^{2} - 4 x) + 6 \cdot - 45$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2} - 4 x - 45)}$

Step 4.2

Factor $x^{2} - 4 x - 45$ using the AC method.

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Step 4.2.1

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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Step 4.2.2

Write the factored form using these integers.

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 ((x - 9) (x + 5))}$

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x - 9) (x + 5)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{2 x^{3} (x + 5) (x - 5) ((x + 9) (x - 9))}{(x - 5) (x + 9) (6 (x - 9) (x + 5))}$

Step 5.2

Cancel the common factor of $2$ and $6$ .

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Step 5.2.1

Factor $2$ out of $2 x^{3} (x + 5) (x - 5) ((x + 9) (x - 9))$ .

$\frac{2 (((x^{3} (x + 5)) (x - 5)) ((x + 9) (x - 9)))}{(x - 5) (x + 9) (6 (x - 9) (x + 5))}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $(x - 5) (x + 9) (6 (x - 9) (x + 5))$ .

$\frac{2 (((x^{3} (x + 5)) (x - 5)) ((x + 9) (x - 9)))}{2 ((x - 5) (x + 9) (3 (x - 9) (x + 5)))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 (((x^{3} (x + 5)) (x - 5)) ((x + 9) (x - 9)))}{2 ((x - 5) (x + 9) (3 (x - 9) (x + 5)))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{((x^{3} (x + 5)) (x - 5)) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9) (x + 5))}$

Step 5.3

Cancel the common factor of $x + 5$ .

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Step 5.3.1

Cancel the common factor.

$\frac{x^{3} (x + 5) (x - 5) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9) (x + 5))}$

Step 5.3.2

Rewrite the expression.

$\frac{((x^{3}) (x - 5)) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9))}$

Step 5.4

Cancel the common factor of $x - 5$ .

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Step 5.4.1

Cancel the common factor.

$\frac{x^{3} (x - 5) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9))}$

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Cancel the common factor of $x + 9$ .

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Step 5.5.1

Cancel the common factor.

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

Step 5.5.2

Rewrite the expression.

$\frac{(x^{3}) (x - 9)}{3 (x - 9)}$

Step 5.6

Cancel the common factor of $x - 9$ .

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Step 5.6.1

Cancel the common factor.

$\frac{x^{3} (x - 9)}{3 (x - 9)}$

Step 5.6.2

Rewrite the expression.

$\frac{x^{3}}{3}$