Algebra Examples

$\frac{a^{2} - 5}{a^{3} - 1} - \frac{a + 1}{a^{2} + a + 1}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{a^{2} - 5}{a^{3} - 1^{3}} - \frac{a + 1}{a^{2} + a + 1}$

Step 1.2

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

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a \cdot 1 + 1^{2})} - \frac{a + 1}{a^{2} + a + 1}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $a$ by $1$ .

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a + 1^{2})} - \frac{a + 1}{a^{2} + a + 1}$

Step 1.3.2

One to any power is one.

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a + 1)} - \frac{a + 1}{a^{2} + a + 1}$

Step 2

To write $- \frac{a + 1}{a^{2} + a + 1}$ as a fraction with a common denominator, multiply by $\frac{a - 1}{a - 1}$ .

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a + 1)} - \frac{a + 1}{a^{2} + a + 1} \cdot \frac{a - 1}{a - 1}$

Step 3

Write each expression with a common denominator of $(a - 1) (a^{2} + a + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{a + 1}{a^{2} + a + 1}$ by $\frac{a - 1}{a - 1}$ .

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a + 1)} - \frac{(a + 1) (a - 1)}{(a^{2} + a + 1) (a - 1)}$

Step 3.2

Reorder the factors of $(a^{2} + a + 1) (a - 1)$ .

$\frac{a^{2} - 5}{(a - 1) (a^{2} + a + 1)} - \frac{(a + 1) (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{a^{2} - 5 - (a + 1) (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{a^{2} - 5 + (- a - 1 \cdot 1) (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 5.2

Multiply $- 1$ by $1$ .

$\frac{a^{2} - 5 + (- a - 1) (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 5.3

Expand $(- a - 1) (a - 1)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{a^{2} - 5 - a (a - 1) - 1 (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 5.3.2

Apply the distributive property.

$\frac{a^{2} - 5 - a \cdot a - a \cdot - 1 - 1 (a - 1)}{(a - 1) (a^{2} + a + 1)}$

Step 5.3.3

Apply the distributive property.

$\frac{a^{2} - 5 - a \cdot a - a \cdot - 1 - 1 a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 5.4.1.1.1

Move $a$ .

$\frac{a^{2} - 5 - (a \cdot a) - a \cdot - 1 - 1 a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.1.1.2

Multiply $a$ by $a$ .

$\frac{a^{2} - 5 - a^{2} - a \cdot - 1 - 1 a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.1.2

Multiply $- a \cdot - 1$ .

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Step 5.4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} - 5 - a^{2} + 1 a - 1 a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.1.2.2

Multiply $a$ by $1$ .

$\frac{a^{2} - 5 - a^{2} + a - 1 a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.1.3

Rewrite $- 1 a$ as $- a$ .

$\frac{a^{2} - 5 - a^{2} + a - a - 1 \cdot - 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.1.4

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} - 5 - a^{2} + a - a + 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.2

Subtract $a$ from $a$ .

$\frac{a^{2} - 5 - a^{2} + 0 + 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.4.3

Add $- a^{2}$ and $0$ .

$\frac{a^{2} - 5 - a^{2} + 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.5

Subtract $a^{2}$ from $a^{2}$ .

$\frac{0 - 5 + 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.6

Subtract $5$ from $0$ .

$\frac{- 5 + 1}{(a - 1) (a^{2} + a + 1)}$

Step 5.7

Add $- 5$ and $1$ .

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

Step 6

Move the negative in front of the fraction.

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