Algebra Examples

$\frac{a^{2} + 6 a - 7}{2 a^{3} + 4 a^{2} + 2 a} \cdot \frac{2 a}{a^{2} - 1}$

Step 1

Factor $a^{2} + 6 a - 7$ 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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 1.2

Write the factored form using these integers.

$\frac{(a - 1) (a + 7)}{2 a^{3} + 4 a^{2} + 2 a} \cdot \frac{2 a}{a^{2} - 1}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 a$ out of $2 a^{3} + 4 a^{2} + 2 a$ .

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Step 2.1.1

Factor $2 a$ out of $2 a^{3}$ .

$\frac{(a - 1) (a + 7)}{2 a (a^{2}) + 4 a^{2} + 2 a} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.1.2

Factor $2 a$ out of $4 a^{2}$ .

$\frac{(a - 1) (a + 7)}{2 a (a^{2}) + 2 a (2 a) + 2 a} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.1.3

Factor $2 a$ out of $2 a$ .

$\frac{(a - 1) (a + 7)}{2 a (a^{2}) + 2 a (2 a) + 2 a (1)} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.1.4

Factor $2 a$ out of $2 a (a^{2}) + 2 a (2 a)$ .

$\frac{(a - 1) (a + 7)}{2 a (a^{2} + 2 a) + 2 a (1)} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.1.5

Factor $2 a$ out of $2 a (a^{2} + 2 a) + 2 a (1)$ .

$\frac{(a - 1) (a + 7)}{2 a (a^{2} + 2 a + 1)} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{(a - 1) (a + 7)}{2 a (a^{2} + 2 a + 1^{2})} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.2.2

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

$2 a = 2 \cdot a \cdot 1$

Step 2.2.3

Rewrite the polynomial.

$\frac{(a - 1) (a + 7)}{2 a (a^{2} + 2 \cdot a \cdot 1 + 1^{2})} \cdot \frac{2 a}{a^{2} - 1}$

Step 2.2.4

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

$\frac{(a - 1) (a + 7)}{2 a {(a + 1)}^{2}} \cdot \frac{2 a}{a^{2} - 1}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(a - 1) (a + 7)}{2 a {(a + 1)}^{2}} \cdot \frac{2 a}{a^{2} - 1^{2}}$

Step 3.2

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

$\frac{(a - 1) (a + 7)}{2 a {(a + 1)}^{2}} \cdot \frac{2 a}{(a + 1) (a - 1)}$

Step 4

Combine.

$\frac{(a - 1) (a + 7) (2 a)}{2 a {(a + 1)}^{2} ((a + 1) (a - 1))}$

Step 5

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

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Step 5.1

Move $a + 1$ .

$\frac{(a - 1) (a + 7) (2 a)}{2 a ((a + 1) {(a + 1)}^{2}) (a - 1)}$

Step 5.2

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

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Step 5.2.1

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

$\frac{(a - 1) (a + 7) (2 a)}{2 a ({(a + 1)}^{1} {(a + 1)}^{2}) (a - 1)}$

Step 5.2.2

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

$\frac{(a - 1) (a + 7) (2 a)}{2 a {(a + 1)}^{1 + 2} (a - 1)}$

Step 5.3

Add $1$ and $2$ .

$\frac{(a - 1) (a + 7) (2 a)}{2 a {(a + 1)}^{3} (a - 1)}$

Step 6

Cancel the common factor of $a - 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(a - 1) (a + 7) (2 a)}{2 a {(a + 1)}^{3} (a - 1)}$

Step 6.2

Rewrite the expression.

$\frac{(a + 7) (2 a)}{2 a {(a + 1)}^{3}}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{(a + 7) (2 a)}{2 a {(a + 1)}^{3}}$

Step 7.2

Rewrite the expression.

$\frac{(a + 7) (a)}{a {(a + 1)}^{3}}$

Step 8

Cancel the common factor of $a$ .

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Step 8.1

Cancel the common factor.

$\frac{(a + 7) a}{a {(a + 1)}^{3}}$

Step 8.2

Rewrite the expression.

$\frac{a + 7}{{(a + 1)}^{3}}$