Algebra Examples

$\frac{x^{2} - 6 x + 8}{2 - x} \cdot \frac{x^{2} + x - 2}{3 x^{3} - 48 x} \div \frac{1 - x^{2}}{- x - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 6 x + 8}{2 - x} \cdot \frac{x^{2} + x - 2}{3 x^{3} - 48 x} \frac{- x - 1}{1 - x^{2}}$

Step 2

Factor $x^{2} - 6 x + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{x^{2} + x - 2}{3 x^{3} - 48 x} \frac{- x - 1}{1 - x^{2}}$

Step 3

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

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Step 3.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x^{3} - 48 x} \frac{- x - 1}{1 - x^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $3 x$ out of $3 x^{3} - 48 x$ .

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Step 4.1.1

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

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x^{2}) - 48 x} \frac{- x - 1}{1 - x^{2}}$

Step 4.1.2

Factor $3 x$ out of $- 48 x$ .

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x^{2}) + 3 x (- 16)} \frac{- x - 1}{1 - x^{2}}$

Step 4.1.3

Factor $3 x$ out of $3 x (x^{2}) + 3 x (- 16)$ .

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x^{2} - 16)} \frac{- x - 1}{1 - x^{2}}$

Step 4.2

Rewrite $16$ as $4^{2}$ .

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x^{2} - 4^{2})} \frac{- x - 1}{1 - x^{2}}$

Step 4.3

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 $b = 4$ .

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x + 4) (x - 4)} \frac{- x - 1}{1 - x^{2}}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 4$ .

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Step 5.1.1

Factor $x - 4$ out of $3 x (x + 4) (x - 4)$ .

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{(x - 4) (3 x (x + 4))} \frac{- x - 1}{1 - x^{2}}$

Step 5.1.2

Cancel the common factor.

$\frac{(x - 4) (x - 2)}{2 - x} \cdot \frac{(x - 1) (x + 2)}{(x - 4) (3 x (x + 4))} \frac{- x - 1}{1 - x^{2}}$

Step 5.1.3

Rewrite the expression.

$\frac{x - 2}{2 - x} \cdot \frac{(x - 1) (x + 2)}{3 x (x + 4)} \frac{- x - 1}{1 - x^{2}}$

Step 5.2

Multiply $\frac{x - 2}{2 - x}$ by $\frac{(x - 1) (x + 2)}{3 x (x + 4)}$ .

$\frac{(x - 2) ((x - 1) (x + 2))}{(2 - x) (3 x (x + 4))} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3

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

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Step 5.3.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) - 2) ((x - 1) (x + 2))}{(2 - x) (3 x (x + 4))} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- 1 (- x) - 1 (2)) ((x - 1) (x + 2))}{(2 - x) (3 x (x + 4))} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

$\frac{- 1 (- x + 2) ((x - 1) (x + 2))}{(2 - x) (3 x (x + 4))} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3.4

Reorder terms.

$\frac{- 1 (- x + 2) ((x - 1) (x + 2))}{(- x + 2) \cdot 3 x (x + 4)} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3.5

Cancel the common factor.

$\frac{- 1 (- x + 2) ((x - 1) (x + 2))}{(- x + 2) \cdot 3 x (x + 4)} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.3.6

Rewrite the expression.

$\frac{- 1 ((x - 1) (x + 2))}{3 x (x + 4)} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- x - 1}{1 - x^{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- x - 1}{1^{2} - x^{2}}$

Step 6.2

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 $b = x$ .

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- x - 1}{(1 + x) (1 - x)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

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

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- (x) - 1}{(1 + x) (1 - x)}$

Step 7.1.2

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

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- (x) - 1 (1)}{(1 + x) (1 - x)}$

Step 7.1.3

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

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- (x + 1)}{(1 + x) (1 - x)}$

Step 7.1.4

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

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- 1 (x + 1)}{(1 + x) (1 - x)}$

Step 7.1.5

Reorder terms.

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- 1 (x + 1)}{(x + 1) (1 - x)}$

Step 7.1.6

Cancel the common factor.

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- 1 (x + 1)}{(x + 1) (1 - x)}$

Step 7.1.7

Rewrite the expression.

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{- 1}{1 - x}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{(x - 1) (x + 2)}{3 x (x + 4)} (- \frac{1}{1 - x})$

Step 8

Multiply $- \frac{(x - 1) (x + 2)}{3 x (x + 4)} (- \frac{1}{1 - x})$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$1 \frac{(x - 1) (x + 2)}{3 x (x + 4)} \frac{1}{1 - x}$

Step 8.2

Multiply $\frac{(x - 1) (x + 2)}{3 x (x + 4)}$ by $1$ .

$\frac{(x - 1) (x + 2)}{3 x (x + 4)} \cdot \frac{1}{1 - x}$

Step 8.3

Multiply $\frac{(x - 1) (x + 2)}{3 x (x + 4)}$ by $\frac{1}{1 - x}$ .

$\frac{(x - 1) (x + 2)}{3 x (x + 4) (1 - x)}$

Step 9

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

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Step 9.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) - 1) (x + 2)}{3 x (x + 4) (1 - x)}$

Step 9.2

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

$\frac{(- 1 (- x) - 1 (1)) (x + 2)}{3 x (x + 4) (1 - x)}$

Step 9.3

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

$\frac{- 1 (- x + 1) (x + 2)}{3 x (x + 4) (1 - x)}$

Step 9.4

Reorder terms.

$\frac{- 1 (- x + 1) (x + 2)}{3 x (x + 4) (- x + 1)}$

Step 9.5

Cancel the common factor.

$\frac{- 1 (- x + 1) (x + 2)}{3 x (x + 4) (- x + 1)}$

Step 9.6

Rewrite the expression.

$\frac{- 1 (x + 2)}{3 x (x + 4)}$

Step 10

Move the negative in front of the fraction.

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