Algebra Examples

$\frac{x^{2} + 2 x + 1}{x^{2} - 3 x - 18} \div \frac{x^{2} - 1}{x^{2} - 7 x + 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 2 x + 1}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 7 x + 6}{x^{2} - 1}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{x^{2} + 2 x + 1^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 7 x + 6}{x^{2} - 1}$

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.

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

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 7 x + 6}{x^{2} - 1}$

Step 2.4

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

$\frac{{(x + 1)}^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 7 x + 6}{x^{2} - 1}$

Step 3

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 3.1

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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 3.2

Write the factored form using these integers.

$\frac{{(x + 1)}^{2}}{(x - 6) (x + 3)} \cdot \frac{x^{2} - 7 x + 6}{x^{2} - 1}$

Step 4

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 4.1

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 $- 7$ .

$- 6, - 1$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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 $b = 1$ .

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

Step 6

Combine.

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

Step 7

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 7.1

Factor $x + 1$ out of ${(x + 1)}^{2} ((x - 6) (x - 1))$ .

$\frac{(x + 1) ((x + 1) ((x - 6) (x - 1)))}{(x - 6) (x + 3) ((x + 1) (x - 1))}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x + 1$ out of $(x - 6) (x + 3) ((x + 1) (x - 1))$ .

$\frac{(x + 1) ((x + 1) ((x - 6) (x - 1)))}{(x + 1) ((x - 6) (x + 3) (x - 1))}$

Step 7.2.2

Cancel the common factor.

$\frac{(x + 1) ((x + 1) ((x - 6) (x - 1)))}{(x + 1) ((x - 6) (x + 3) (x - 1))}$

Step 7.2.3

Rewrite the expression.

$\frac{(x + 1) ((x - 6) (x - 1))}{(x - 6) (x + 3) (x - 1)}$

Step 8

Cancel the common factor of $x - 6$ .

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Step 8.1

Cancel the common factor.

$\frac{(x + 1) ((x - 6) (x - 1))}{(x - 6) (x + 3) (x - 1)}$

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $x - 1$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

$\frac{x + 1}{x + 3}$