Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 2

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

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Step 2.1

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 $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2

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

$2 x = 2 \cdot x \cdot 1$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 1$ .

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x + 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $x + 1$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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