Algebra Examples

$\frac{x + 3}{x - 1} \cdot \frac{x^{2} - 2 x + 1}{x^{2} + 5 x + 6}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{x + 3}{x - 1} \cdot \frac{x^{2} - 2 x + 1^{2}}{x^{2} + 5 x + 6}$

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.

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

Step 1.3

Rewrite the polynomial.

$\frac{x + 3}{x - 1} \cdot \frac{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}{x^{2} + 5 x + 6}$

Step 1.4

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

$\frac{x + 3}{x - 1} \cdot \frac{{(x - 1)}^{2}}{x^{2} + 5 x + 6}$

Step 2

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Step 2.2

Write the factored form using these integers.

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Factor $x + 3$ out of $(x + 2) (x + 3)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 1$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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