Algebra Examples

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

Step 1

Factor $4$ out of $4 x - 8$ .

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Step 1.1

Factor $4$ out of $4 x$ .

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

Step 1.2

Factor $4$ out of $- 8$ .

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

Step 1.3

Factor $4$ out of $4 x + 4 \cdot - 2$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\frac{4}{x - 2}$

Step 4

Since no factors can be removed from the denominator, there are no holes in the graph.

No holes

Step 5