Simplify (1-a)/(a^2-a+1)+(a^2)/(a^3+1)

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

Simplify.

Multiply $- 1$ by $1$ .

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

One to any power is one.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $1$ .

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

Multiply $1$ by $1$ .

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

Multiply $- 1$ by $1$ .

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

Subtract $a$ from $a$ .

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

Add $1$ and $0$ .

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

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

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

Add $0$ and $1$ .

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

Simplify (1-a)/(a^2-a+1)+(a^2)/(a^3+1)

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