Algebra Examples

$\frac{a^{3} + 1}{a^{4} - a^{3} + a - 1}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{a^{3} + 1^{3}}{a^{4} - a^{3} + a - 1}$

Step 1.2

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 $b = 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 1$ by $1$ .

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

Step 1.3.2

One to any power is one.

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

Step 2

Simplify the denominator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

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

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $a - 1$ .

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

Step 2.3

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

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

Step 2.4

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 $b = 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 1$ by $1$ .

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

Step 2.5.2

One to any power is one.

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $a + 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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