Solve frac{1-a^{3}}{a^{2}-2a+1}-a^2-1/a^3-3a^2+3a-1+a^2/a-1

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{1-a^{3}}{a^{2}-2a+1}.

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

Cancel out a-1 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}-1}{a^{3}-3a^{2}+3a-1}.

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and \left(a-1\right)^{2} is \left(a-1\right)^{2}. Multiply \frac{-a^{2}-a-1}{a-1} times \frac{a-1}{a-1}.

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

Since \frac{\left(-a^{2}-a-1\right)\left(a-1\right)}{\left(a-1\right)^{2}} and \frac{a+1}{\left(a-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(-a^{2}-a-1\right)\left(a-1\right)-\left(a+1\right).

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

Combine like terms in -a^{3}+a^{2}-a^{2}+a-a+1-a-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)^{2} and a-1 is \left(a-1\right)^{2}. Multiply \frac{a^{2}}{a-1} times \frac{a-1}{a-1}.

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

Since \frac{-a^{3}-a}{\left(a-1\right)^{2}} and \frac{a^{2}\left(a-1\right)}{\left(a-1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{3}-a+a^{2}\left(a-1\right).

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

Combine like terms in -a^{3}-a+a^{3}-a^{2}.

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

Expand \left(a-1\right)^{2}.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{1-a^{3}}{a^{2}-2a+1}.

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

Cancel out a-1 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}-1}{a^{3}-3a^{2}+3a-1}.

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and \left(a-1\right)^{2} is \left(a-1\right)^{2}. Multiply \frac{-a^{2}-a-1}{a-1} times \frac{a-1}{a-1}.

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

Since \frac{\left(-a^{2}-a-1\right)\left(a-1\right)}{\left(a-1\right)^{2}} and \frac{a+1}{\left(a-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(-a^{2}-a-1\right)\left(a-1\right)-\left(a+1\right).

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

Combine like terms in -a^{3}+a^{2}-a^{2}+a-a+1-a-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)^{2} and a-1 is \left(a-1\right)^{2}. Multiply \frac{a^{2}}{a-1} times \frac{a-1}{a-1}.

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

Since \frac{-a^{3}-a}{\left(a-1\right)^{2}} and \frac{a^{2}\left(a-1\right)}{\left(a-1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{3}-a+a^{2}\left(a-1\right).

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

Combine like terms in -a^{3}-a+a^{3}-a^{2}.

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

Expand \left(a-1\right)^{2}.

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