Algebra Examples

$\frac{8 x - 40}{40 - 3 x - x^{2}} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 1

Factor $8$ out of $8 x - 40$ .

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Step 1.1

Factor $8$ out of $8 x$ .

$\frac{8 (x) - 40}{40 - 3 x - x^{2}} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 1.2

Factor $8$ out of $- 40$ .

$\frac{8 x + 8 \cdot - 5}{40 - 3 x - x^{2}} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 1.3

Factor $8$ out of $8 x + 8 \cdot - 5$ .

$\frac{8 (x - 5)}{40 - 3 x - x^{2}} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$\frac{8 (x - 5)}{- x^{2} - 3 x + 40} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 40 = - 40$ and whose sum is $b = - 3$ .

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Step 2.2.1

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

$\frac{8 (x - 5)}{- x^{2} - 3 (x) + 40} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.2.2

Rewrite $- 3$ as $5$ plus $- 8$

$\frac{8 (x - 5)}{- x^{2} + (5 - 8) x + 40} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.2.3

Apply the distributive property.

$\frac{8 (x - 5)}{- x^{2} + 5 x - 8 x + 40} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$\frac{8 (x - 5)}{(- x^{2} + 5 x) - 8 x + 40} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{8 (x - 5)}{x (- x + 5) + 8 (- x + 5)} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $- x + 5$ .

$\frac{8 (x - 5)}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 x^{2} - 8 x}$

Step 3

Factor $2 x$ out of $2 x^{2} - 8 x$ .

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Step 3.1

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

$\frac{8 (x - 5)}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 x (x) - 8 x}$

Step 3.2

Factor $2 x$ out of $- 8 x$ .

$\frac{8 (x - 5)}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 x (x) + 2 x (- 4)}$

Step 3.3

Factor $2 x$ out of $2 x (x) + 2 x (- 4)$ .

$\frac{8 (x - 5)}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 x (x - 4)}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $8 (x - 5)$ .

$\frac{2 (4 (x - 5))}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 x (x - 4)}$

Step 4.2

Factor $2$ out of $2 x (x - 4)$ .

$\frac{2 (4 (x - 5))}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 (x (x - 4))}$

Step 4.3

Cancel the common factor.

$\frac{2 (4 (x - 5))}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{2 (x (x - 4))}$

Step 4.4

Rewrite the expression.

$\frac{4 (x - 5)}{(- x + 5) (x + 8)} \cdot \frac{x - 8}{x (x - 4)}$

Step 5

Multiply $\frac{4 (x - 5)}{(- x + 5) (x + 8)}$ by $\frac{x - 8}{x (x - 4)}$ .

$\frac{4 (x - 5) (x - 8)}{(- x + 5) (x + 8) (x (x - 4))}$

Step 6

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

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Step 6.1

Factor $- 1$ out of $x$ .

$\frac{4 (- 1 (- x) - 5) (x - 8)}{(- x + 5) (x + 8) (x (x - 4))}$

Step 6.2

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

$\frac{4 (- 1 (- x) - 1 (5)) (x - 8)}{(- x + 5) (x + 8) (x (x - 4))}$

Step 6.3

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

$\frac{4 (- 1 (- x + 5)) (x - 8)}{(- x + 5) (x + 8) (x (x - 4))}$

Step 6.4

Cancel the common factor.

$\frac{4 (- 1 (- x + 5)) (x - 8)}{(- x + 5) (x + 8) (x (x - 4))}$

Step 6.5

Rewrite the expression.

$\frac{4 \cdot (- 1) (x - 8)}{(x + 8) (x (x - 4))}$

Step 7

Simplify the expression.

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Step 7.1

Multiply $4$ by $- 1$ .

$\frac{- 4 (x - 8)}{(x + 8) x (x - 4)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{4 (x - 8)}{(x + 8) x (x - 4)}$

Step 7.3

Reorder factors in $- \frac{4 (x - 8)}{(x + 8) x (x - 4)}$ .

$- \frac{4 (x - 8)}{x (x + 8) (x - 4)}$