Solve (9v-6)/(v^{2-6v+5)}-(5v^3-5v^2/(5v^5-10v^2)

$$\frac { ( 9 v - 6 ) } { ( v ^ { 2 } - 6 v + 5 ) } - \frac { ( 5 v ^ { 3 } - 5 v ^ { 2 } } { ( 5 v ^ { 5 } - 10 v ^ { 2 } ) }$$

Evaluate

$$\frac{9v^{4}-7v^{3}+7v^{2}-29v+17}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{5v^{3}-5v^{2}}{5v^{5}-10v^{2}}.

$$\frac{9v-6}{v^{2}-6v+5}-\frac{5\left(v-1\right)v^{2}}{5v^{2}\left(v^{3}-2\right)}$$

Cancel out 5v^{2} in both numerator and denominator.

$$\frac{9v-6}{v^{2}-6v+5}-\frac{v-1}{v^{3}-2}$$

Factor v^{2}-6v+5.

$$\frac{9v-6}{\left(v-5\right)\left(v-1\right)}-\frac{v-1}{v^{3}-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-5\right)\left(v-1\right) and v^{3}-2 is \left(v-5\right)\left(v-1\right)\left(v^{3}-2\right). Multiply \frac{9v-6}{\left(v-5\right)\left(v-1\right)} times \frac{v^{3}-2}{v^{3}-2}. Multiply \frac{v-1}{v^{3}-2} times \frac{\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)}.

$$\frac{\left(9v-6\right)\left(v^{3}-2\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}-\frac{\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Since \frac{\left(9v-6\right)\left(v^{3}-2\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)} and \frac{\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(9v-6\right)\left(v^{3}-2\right)-\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Do the multiplications in \left(9v-6\right)\left(v^{3}-2\right)-\left(v-1\right)\left(v-5\right)\left(v-1\right).

$$\frac{9v^{4}-18v-6v^{3}+12-v^{3}+6v^{2}-5v+v^{2}-6v+5}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Combine like terms in 9v^{4}-18v-6v^{3}+12-v^{3}+6v^{2}-5v+v^{2}-6v+5.

$$\frac{9v^{4}-29v-7v^{3}+17+7v^{2}}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Expand \left(v-5\right)\left(v-1\right)\left(v^{3}-2\right).

$$\frac{9v^{4}-29v-7v^{3}+17+7v^{2}}{v^{5}-6v^{4}+5v^{3}-2v^{2}+12v-10}$$

Expand

$$\frac{9v^{4}-7v^{3}+7v^{2}-29v+17}{\left(v^{2}-6v+5\right)\left(v^{3}-2\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{5v^{3}-5v^{2}}{5v^{5}-10v^{2}}.

$$\frac{9v-6}{v^{2}-6v+5}-\frac{5\left(v-1\right)v^{2}}{5v^{2}\left(v^{3}-2\right)}$$

Cancel out 5v^{2} in both numerator and denominator.

$$\frac{9v-6}{v^{2}-6v+5}-\frac{v-1}{v^{3}-2}$$

Factor v^{2}-6v+5.

$$\frac{9v-6}{\left(v-5\right)\left(v-1\right)}-\frac{v-1}{v^{3}-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-5\right)\left(v-1\right) and v^{3}-2 is \left(v-5\right)\left(v-1\right)\left(v^{3}-2\right). Multiply \frac{9v-6}{\left(v-5\right)\left(v-1\right)} times \frac{v^{3}-2}{v^{3}-2}. Multiply \frac{v-1}{v^{3}-2} times \frac{\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)}.

$$\frac{\left(9v-6\right)\left(v^{3}-2\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}-\frac{\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Since \frac{\left(9v-6\right)\left(v^{3}-2\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)} and \frac{\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(9v-6\right)\left(v^{3}-2\right)-\left(v-1\right)\left(v-5\right)\left(v-1\right)}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Do the multiplications in \left(9v-6\right)\left(v^{3}-2\right)-\left(v-1\right)\left(v-5\right)\left(v-1\right).

$$\frac{9v^{4}-18v-6v^{3}+12-v^{3}+6v^{2}-5v+v^{2}-6v+5}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Combine like terms in 9v^{4}-18v-6v^{3}+12-v^{3}+6v^{2}-5v+v^{2}-6v+5.

$$\frac{9v^{4}-29v-7v^{3}+17+7v^{2}}{\left(v-5\right)\left(v-1\right)\left(v^{3}-2\right)}$$

Expand \left(v-5\right)\left(v-1\right)\left(v^{3}-2\right).

$$\frac{9v^{4}-29v-7v^{3}+17+7v^{2}}{v^{5}-6v^{4}+5v^{3}-2v^{2}+12v-10}$$