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

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

To divide by a fraction, multiply by its reciprocal.

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

Simplify each term.

Simplify the denominator.

Rewrite ${(a - 1)}^{2}$ as $(a - 1) (a - 1)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Move $- 1$ to the left of $a$ .

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $a$ from $- a$ .

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

Subtract $2 a$ from $3 a$ .

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

Simplify.

Multiply $a$ by $1$ .

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

One to any power is one.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Move $- 1$ to the left of $a$ .

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $a$ from $- a$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 1$ by $1$ .

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

Multiply $- 3$ by $- 1$ .

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

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

$(\frac{- 2 a + 1 - 1 + 3 a + 0}{(a - 1) (a^{2} + a + 1)} - \frac{1}{a - 1}) \frac{a - 1}{a^{2} + 1}$

Add $- 2 a + 1 - 1 + 3 a$ and $0$ .

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

Add $- 2 a$ and $3 a$ .

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

Subtract $1$ from $1$ .

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

Add $a$ and $0$ .

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $a$ from $a$ .

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

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

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

Simplify terms.

Cancel the common factor of $a - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Rewrite $- 1$ as $- 1 (1)$ .

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

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

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

Rewrite $- (a^{2} + 1)$ as $- 1 (a^{2} + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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