Algebra Examples

$\frac{4}{- x - 1} \cdot \frac{x^{2} - 11 x + 30}{5 - x} \div \frac{x^{2} - 7 x + 6}{1 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4}{- x - 1} \cdot \frac{x^{2} - 11 x + 30}{5 - x} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 2

Factor $x^{2} - 11 x + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 2.2

Write the factored form using these integers.

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (x - 5)}{5 - x} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x - 5$ and $5 - x$ .

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Step 3.1.1

Factor $- 1$ out of $x$ .

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (- 1 (- x) - 5)}{5 - x} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (- 1 (- x) - 1 (5))}{5 - x} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (- 1 (- x + 5))}{5 - x} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.1.4

Reorder terms.

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (- 1 (- x + 5))}{- x + 5} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.1.5

Cancel the common factor.

$\frac{4}{- x - 1} \cdot \frac{(x - 6) (- 1 (- x + 5))}{- x + 5} \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.1.6

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

$\frac{4}{- x - 1} \cdot ((x - 6) \cdot (- 1)) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.2

Rewrite using the commutative property of multiplication.

$- \frac{4}{- x - 1} (x - 6) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.3

Apply the distributive property.

$(- \frac{4}{- x - 1} x - \frac{4}{- x - 1} \cdot - 6) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 3.4

Combine $x$ and $\frac{4}{- x - 1}$ .

$(- \frac{x \cdot 4}{- x - 1} - \frac{4}{- x - 1} \cdot - 6) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 4

Multiply $- \frac{4}{- x - 1} \cdot - 6$ .

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Step 4.1

Multiply $- 6$ by $- 1$ .

$(- \frac{x \cdot 4}{- x - 1} + 6 \frac{4}{- x - 1}) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 4.2

Combine $6$ and $\frac{4}{- x - 1}$ .

$(- \frac{x \cdot 4}{- x - 1} + \frac{6 \cdot 4}{- x - 1}) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 4.3

Multiply $6$ by $4$ .

$(- \frac{x \cdot 4}{- x - 1} + \frac{24}{- x - 1}) \frac{1 - x^{2}}{x^{2} - 7 x + 6}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$(- \frac{x \cdot 4}{- x - 1} + \frac{24}{- x - 1}) \frac{1^{2} - x^{2}}{x^{2} - 7 x + 6}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$(- \frac{x \cdot 4}{- x - 1} + \frac{24}{- x - 1}) \frac{(1 + x) (1 - x)}{x^{2} - 7 x + 6}$

Step 6

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 6.1

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 $- 7$ .

$- 6, - 1$

Step 6.2

Write the factored form using these integers.

$(- \frac{x \cdot 4}{- x - 1} + \frac{24}{- x - 1}) \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7

Simplify terms.

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Step 7.1

Combine the numerators over the common denominator.

$\frac{- x \cdot 4 + 24}{- x - 1} \cdot \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7.2

Multiply $4$ by $- 1$ .

$\frac{- 4 x + 24}{- x - 1} \cdot \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7.3

Factor $4$ out of $- 4 x + 24$ .

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Step 7.3.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 24}{- x - 1} \cdot \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7.3.2

Factor $4$ out of $24$ .

$\frac{4 (- x) + 4 (6)}{- x - 1} \cdot \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7.3.3

Factor $4$ out of $4 (- x) + 4 (6)$ .

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (1 - x)}{(x - 6) (x - 1)}$

Step 7.4

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 7.4.1

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

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (- 1 (- 1) - x)}{(x - 6) (x - 1)}$

Step 7.4.2

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

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (- 1 (- 1) - (x))}{(x - 6) (x - 1)}$

Step 7.4.3

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

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (- 1 (- 1 + x))}{(x - 6) (x - 1)}$

Step 7.4.4

Reorder terms.

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (- 1 (x - 1))}{(x - 6) (x - 1)}$

Step 7.4.5

Cancel the common factor.

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) (- 1 (x - 1))}{(x - 6) (x - 1)}$

Step 7.4.6

Rewrite the expression.

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{(1 + x) \cdot (- 1)}{x - 6}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Move $- 1$ to the left of $1 + x$ .

$\frac{4 (- x + 6)}{- x - 1} \cdot \frac{- 1 \cdot (1 + x)}{x - 6}$

Step 7.5.2

Move the negative in front of the fraction.

$\frac{4 (- x + 6)}{- x - 1} (- \frac{1 + x}{x - 6})$

Step 7.5.3

Rewrite using the commutative property of multiplication.

$- \frac{4 (- x + 6)}{- x - 1} \cdot \frac{1 + x}{x - 6}$

Step 7.6

Multiply $\frac{1 + x}{x - 6}$ by $\frac{4 (- x + 6)}{- x - 1}$ .

$- \frac{(1 + x) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7

Cancel the common factor of $1 + x$ and $- x - 1$ .

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Step 7.7.1

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

$- \frac{(- 1 (- 1) + x) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7.2

Factor $- 1$ out of $x$ .

$- \frac{(- 1 (- 1) - 1 (- x)) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7.3

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

$- \frac{- 1 (- 1 - x) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7.4

Reorder terms.

$- \frac{- 1 (- x - 1) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7.5

Cancel the common factor.

$- \frac{- 1 (- x - 1) (4 (- x + 6))}{(x - 6) (- x - 1)}$

Step 7.7.6

Rewrite the expression.

$- \frac{- 1 (4 (- x + 6))}{x - 6}$

Step 7.8

Cancel the common factor of $- x + 6$ and $x - 6$ .

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Step 7.8.1

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

$- \frac{- 1 (4 (- (x) + 6))}{x - 6}$

Step 7.8.2

Rewrite $6$ as $- 1 (- 6)$ .

$- \frac{- 1 (4 (- (x) - 1 (- 6)))}{x - 6}$

Step 7.8.3

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$- \frac{- 1 (4 (- (x - 6)))}{x - 6}$

Step 7.8.4

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$- \frac{- 1 (4 (- 1 (x - 6)))}{x - 6}$

Step 7.8.5

Cancel the common factor.

$- \frac{- 1 \cdot 4 (- 1 (x - 6))}{x - 6}$

Step 7.8.6

Divide $- 1 \cdot 4 \cdot (- 1)$ by $1$ .

$- (- 1 \cdot 4 \cdot (- 1))$

Step 8

Multiply $- (- 1 \cdot 4 \cdot (- 1))$ .

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Step 8.1

Multiply $- 1$ by $4$ .

$- (- 4 \cdot - 1)$

Step 8.2

Multiply $- 4$ by $- 1$ .

$- 1 \cdot 4$

Step 8.3

Multiply $- 1$ by $4$ .

$- 4$