Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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 $a = a$ and $b = 1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{3}$ .

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

Step 2.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = a$ and $b = 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Multiply $- 1$ by $1$ .

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

Step 2.3.2

One to any power is one.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

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.

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

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $a + 1$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $a^{2} - a + 1$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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