Algebra Examples

$\frac{8 - x^{3}}{25 x^{2} + 100 - 100 x}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

$\frac{2^{3} - x^{3}}{25 x^{2} + 100 - 100 x}$

Step 1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2$ and $b = x$ .

$\frac{(2 - x) (2^{2} + 2 x + x^{2})}{25 x^{2} + 100 - 100 x}$

Step 1.3

Raise $2$ to the power of $2$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 x^{2} + 100 - 100 x}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $25$ out of $25 x^{2} + 100 - 100 x$ .

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Step 2.1.1

Factor $25$ out of $25 x^{2}$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2}) + 100 - 100 x}$

Step 2.1.2

Factor $25$ out of $100$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 x^{2} + 25 \cdot 4 - 100 x}$

Step 2.1.3

Factor $25$ out of $- 100 x$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 x^{2} + 25 \cdot 4 + 25 (- 4 x)}$

Step 2.1.4

Factor $25$ out of $25 x^{2} + 25 \cdot 4$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2} + 4) + 25 (- 4 x)}$

Step 2.1.5

Factor $25$ out of $25 (x^{2} + 4) + 25 (- 4 x)$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2} + 4 - 4 x)}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rearrange terms.

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2} - 4 x + 4)}$

Step 2.2.2

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

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2} - 4 x + 2^{2})}$

Step 2.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.2.4

Rewrite the polynomial.

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 (x^{2} - 2 \cdot x \cdot 2 + 2^{2})}$

Step 2.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$\frac{(2 - x) (4 + 2 x + x^{2})}{25 {(x - 2)}^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $2 - x$ and ${(x - 2)}^{2}$ .

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Step 3.1.1

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

$\frac{(- 1 (- 2) - x) (4 + 2 x + x^{2})}{25 {(x - 2)}^{2}}$

Step 3.1.2

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

$\frac{(- 1 (- 2) - (x)) (4 + 2 x + x^{2})}{25 {(x - 2)}^{2}}$

Step 3.1.3

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

$\frac{- 1 (- 2 + x) (4 + 2 x + x^{2})}{25 {(x - 2)}^{2}}$

Step 3.1.4

Reorder terms.

$\frac{- 1 (x - 2) (4 + 2 x + x^{2})}{25 {(x - 2)}^{2}}$

Step 3.1.5

Factor $x - 2$ out of $- 1 (x - 2) (4 + 2 x + x^{2})$ .

$\frac{(x - 2) (- 1 (4 + 2 x + x^{2}))}{25 {(x - 2)}^{2}}$

Step 3.1.6

Cancel the common factors.

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Step 3.1.6.1

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

$\frac{(x - 2) (- 1 (4 + 2 x + x^{2}))}{(x - 2) (25 (x - 2))}$

Step 3.1.6.2

Cancel the common factor.

$\frac{(x - 2) (- 1 (4 + 2 x + x^{2}))}{(x - 2) (25 (x - 2))}$

Step 3.1.6.3

Rewrite the expression.

$\frac{- 1 (4 + 2 x + x^{2})}{25 (x - 2)}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{4 + 2 x + x^{2}}{25 (x - 2)}$