Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 1$ by $1$ .

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

Step 1.3.2

One to any power is one.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $a$ by $1$ .

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

Step 5.2.1.2

Multiply $1$ by $1$ .

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

Step 5.2.1.3

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

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Step 5.2.1.3.1

Move $a$ .

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

Step 5.2.1.3.2

Multiply $a$ by $a$ .

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

Step 5.2.1.4

Multiply $- 1$ by $1$ .

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

Step 5.2.2

Subtract $a$ from $a$ .

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

Step 5.2.3

Add $1$ and $0$ .

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

Step 5.3

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

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

Step 5.4

Add $0$ and $1$ .

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