Algebra Examples

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

Step 1

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 1.1

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 $- 3$ .

$- 2, - 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

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.

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

Step 2.4

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

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

$\frac{x - 1}{x + 1} \cdot \frac{x - 2}{x - 2}$

Step 3.2

Multiply $\frac{x - 1}{x + 1}$ by $\frac{x - 2}{x - 2}$ .

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

Step 3.3

Cancel the common factor of $x - 2$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\frac{x - 1}{x + 1}$