Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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 $b = 1$ .

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

Step 3

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

Tap for more steps...

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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

Reduce the expression $\frac{(x + 1) (x - 1)}{(x - 6) (x - 1)}$ by cancelling the common factors.

Tap for more steps...

Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

Tap for more steps...

Step 5.1

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 5.2

Write the factored form using these integers.

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Raise $x - 6$ to the power of $1$ .

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

Step 7.3

Raise $x - 6$ to the power of $1$ .

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

Step 7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Factor using the perfect square rule.

Tap for more steps...

Step 9.1

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

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

Step 9.2

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 9.3

Rewrite the polynomial.

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

Step 9.4

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

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

Step 10

Multiply ${(x + 1)}^{2}$ by $x + 1$ by adding the exponents.

Tap for more steps...

Step 10.1

Multiply ${(x + 1)}^{2}$ by $x + 1$ .

Tap for more steps...

Step 10.1.1

Raise $x + 1$ to the power of $1$ .

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

Step 10.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.2

Add $2$ and $1$ .

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