Algebra Examples

$\frac{24 x - 6 x^{3}}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \div \frac{x^{2} + 5 x + 6}{x^{2} + 7 x - 30}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{24 x - 6 x^{3}}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $6 x$ out of $24 x - 6 x^{3}$ .

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Step 2.1.1

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

$\frac{6 x (4) - 6 x^{3}}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 2.1.2

Factor $6 x$ out of $- 6 x^{3}$ .

$\frac{6 x (4) + 6 x (- x^{2})}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 2.1.3

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

$\frac{6 x (4 - x^{2})}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 2.2

Rewrite $4$ as $2^{2}$ .

$\frac{6 x (2^{2} - x^{2})}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 2.3

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

$\frac{6 x (2 + x) (2 - x)}{3 x - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 3

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

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Step 3.1

Factor $x$ out of $3 x$ .

$\frac{6 x (2 + x) (2 - x)}{x \cdot 3 - x^{2}} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 3.2

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

$\frac{6 x (2 + x) (2 - x)}{x \cdot 3 + x (- x)} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 3.3

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

$\frac{6 x (2 + x) (2 - x)}{x (3 - x)} \cdot \frac{x^{2} + 2 x - 3}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 4

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 4.2

Write the factored form using these integers.

$\frac{6 x (2 + x) (2 - x)}{x (3 - x)} \cdot \frac{(x - 1) (x + 3)}{x^{2} + 9 x - 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 5

Factor $x^{2} + 9 x - 10$ 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 $- 10$ and whose sum is $9$ .

$- 1, 10$

Step 5.2

Write the factored form using these integers.

$\frac{6 x (2 + x) (2 - x)}{x (3 - x)} \cdot \frac{(x - 1) (x + 3)}{(x - 1) (x + 10)} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x$ .

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Step 6.1.1

Cancel the common factor.

$\frac{6 x (2 + x) (2 - x)}{x (3 - x)} \cdot \frac{(x - 1) (x + 3)}{(x - 1) (x + 10)} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 6.1.2

Rewrite the expression.

$\frac{(6 (2 + x)) (2 - x)}{3 - x} \cdot \frac{(x - 1) (x + 3)}{(x - 1) (x + 10)} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 6.2

Cancel the common factor of $x - 1$ .

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Step 6.2.1

Cancel the common factor.

$\frac{6 (2 + x) (2 - x)}{3 - x} \cdot \frac{(x - 1) (x + 3)}{(x - 1) (x + 10)} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 6.2.2

Rewrite the expression.

$\frac{6 (2 + x) (2 - x)}{3 - x} \cdot \frac{x + 3}{x + 10} \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 6.3

Multiply $\frac{6 (2 + x) (2 - x)}{3 - x}$ by $\frac{x + 3}{x + 10}$ .

$\frac{6 (2 + x) (2 - x) (x + 3)}{(3 - x) (x + 10)} \cdot \frac{x^{2} + 7 x - 30}{x^{2} + 5 x + 6}$

Step 7

Factor $x^{2} + 7 x - 30$ 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 $- 30$ and whose sum is $7$ .

$- 3, 10$

Step 7.2

Write the factored form using these integers.

$\frac{6 (2 + x) (2 - x) (x + 3)}{(3 - x) (x + 10)} \cdot \frac{(x - 3) (x + 10)}{x^{2} + 5 x + 6}$

Step 8

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 8.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 8.2

Write the factored form using these integers.

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

Step 9

Simplify terms.

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Step 9.1

Cancel the common factor of $x + 3$ .

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Step 9.1.1

Factor $x + 3$ out of $6 (2 + x) (2 - x) (x + 3)$ .

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

Step 9.1.2

Factor $x + 3$ out of $(x + 2) (x + 3)$ .

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

Step 9.1.3

Cancel the common factor.

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

Step 9.1.4

Rewrite the expression.

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

Step 9.2

Cancel the common factor of $x + 10$ .

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Step 9.2.1

Factor $x + 10$ out of $(3 - x) (x + 10)$ .

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

Step 9.2.2

Factor $x + 10$ out of $(x - 3) (x + 10)$ .

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

Step 9.2.3

Cancel the common factor.

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

Step 9.2.4

Rewrite the expression.

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

Step 9.3

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

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

Step 9.4

Cancel the common factor of $2 + x$ and $x + 2$ .

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Step 9.4.1

Reorder terms.

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

Step 9.4.2

Cancel the common factor.

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

Step 9.4.3

Rewrite the expression.

$\frac{(6 (2 - x)) (x - 3)}{3 - x}$

Step 9.5

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

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Step 9.5.1

Factor $- 1$ out of $x$ .

$\frac{6 (2 - x) (- 1 (- x) - 3)}{3 - x}$

Step 9.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{6 (2 - x) (- 1 (- x) - 1 (3))}{3 - x}$

Step 9.5.3

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

$\frac{6 (2 - x) (- 1 (- x + 3))}{3 - x}$

Step 9.5.4

Reorder terms.

$\frac{6 (2 - x) (- 1 (- x + 3))}{- x + 3}$

Step 9.5.5

Cancel the common factor.

$\frac{6 (2 - x) (- 1 (- x + 3))}{- x + 3}$

Step 9.5.6

Divide $6 (2 - x) \cdot (- 1)$ by $1$ .

$6 (2 - x) \cdot (- 1)$

Step 9.6

Apply the distributive property.

$(6 \cdot 2 + 6 (- x)) \cdot - 1$

Step 9.7

Multiply.

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Step 9.7.1

Multiply $6$ by $2$ .

$(12 + 6 (- x)) \cdot - 1$

Step 9.7.2

Multiply $- 1$ by $6$ .

$(12 - 6 x) \cdot - 1$

Step 9.8

Apply the distributive property.

$12 \cdot - 1 - 6 x \cdot - 1$

Step 9.9

Multiply.

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Step 9.9.1

Multiply $12$ by $- 1$ .

$- 12 - 6 x \cdot - 1$

Step 9.9.2

Multiply $- 1$ by $- 6$ .

$- 12 + 6 x$