Precalculus Examples

$8 - \frac{x^{3}}{x^{2} - 4 x + 4}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$8 - \frac{x^{3}}{x^{2} - 4 x + 2^{2}}$

Step 1.2

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 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

To write $8$ as a fraction with a common denominator, multiply by $\frac{{(x - 2)}^{2}}{{(x - 2)}^{2}}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 4.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 4.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{8 (x^{2} - 2 x - 2 x + 4) - x^{3}}{{(x - 2)}^{2}}$

Step 4.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{8 (x^{2} - 4 x + 4) - x^{3}}{{(x - 2)}^{2}}$

Step 4.4

Apply the distributive property.

$\frac{8 x^{2} + 8 (- 4 x) + 8 \cdot 4 - x^{3}}{{(x - 2)}^{2}}$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $- 4$ by $8$ .

$\frac{8 x^{2} - 32 x + 8 \cdot 4 - x^{3}}{{(x - 2)}^{2}}$

Step 4.5.2

Multiply $8$ by $4$ .

$\frac{8 x^{2} - 32 x + 32 - x^{3}}{{(x - 2)}^{2}}$

Step 4.6

Reorder terms.

$\frac{- x^{3} + 8 x^{2} - 32 x + 32}{{(x - 2)}^{2}}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- x^{3}$ .

$\frac{- (x^{3}) + 8 x^{2} - 32 x + 32}{{(x - 2)}^{2}}$

Step 5.2

Factor $- 1$ out of $8 x^{2}$ .

$\frac{- (x^{3}) - (- 8 x^{2}) - 32 x + 32}{{(x - 2)}^{2}}$

Step 5.3

Factor $- 1$ out of $- (x^{3}) - (- 8 x^{2})$ .

$\frac{- (x^{3} - 8 x^{2}) - 32 x + 32}{{(x - 2)}^{2}}$

Step 5.4

Factor $- 1$ out of $- 32 x$ .

$\frac{- (x^{3} - 8 x^{2}) - (32 x) + 32}{{(x - 2)}^{2}}$

Step 5.5

Factor $- 1$ out of $- (x^{3} - 8 x^{2}) - (32 x)$ .

$\frac{- (x^{3} - 8 x^{2} + 32 x) + 32}{{(x - 2)}^{2}}$

Step 5.6

Rewrite $32$ as $- 1 (- 32)$ .

$\frac{- (x^{3} - 8 x^{2} + 32 x) - 1 (- 32)}{{(x - 2)}^{2}}$

Step 5.7

Factor $- 1$ out of $- (x^{3} - 8 x^{2} + 32 x) - 1 (- 32)$ .

$\frac{- (x^{3} - 8 x^{2} + 32 x - 32)}{{(x - 2)}^{2}}$

Step 5.8

Simplify the expression.

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Step 5.8.1

Rewrite $- (x^{3} - 8 x^{2} + 32 x - 32)$ as $- 1 (x^{3} - 8 x^{2} + 32 x - 32)$ .

$\frac{- 1 (x^{3} - 8 x^{2} + 32 x - 32)}{{(x - 2)}^{2}}$

Step 5.8.2

Move the negative in front of the fraction.

$- \frac{x^{3} - 8 x^{2} + 32 x - 32}{{(x - 2)}^{2}}$