Solve v+1/v-6+6/3v^2

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

Evaluate

$$\frac{v^{3}+v^{2}+2v-12}{\left(v-6\right)v^{2}}$$

Short Solution Steps

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

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

Since \frac{\left(v+1\right)\times 3v^{2}}{3\left(v-6\right)v^{2}} and \frac{6\left(v-6\right)}{3\left(v-6\right)v^{2}} have the same denominator, add them by adding their numerators.

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

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

$$\frac{3v^{3}+3v^{2}+6v-36}{3\left(v-6\right)v^{2}}$$

Factor the expressions that are not already factored in \frac{3v^{3}+3v^{2}+6v-36}{3\left(v-6\right)v^{2}}.

$$\frac{3\left(v^{3}+v^{2}+2v-12\right)}{3\left(v-6\right)v^{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{v^{3}+v^{2}+2v-12}{\left(v-6\right)v^{2}}$$

Expand \left(v-6\right)v^{2}.

$$\frac{v^{3}+v^{2}+2v-12}{v^{3}-6v^{2}}$$

Expand

$$\frac{v^{3}+v^{2}+2v-12}{\left(v-6\right)v^{2}}$$

Short Solution Steps

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

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

Since \frac{\left(v+1\right)\times 3v^{2}}{3\left(v-6\right)v^{2}} and \frac{6\left(v-6\right)}{3\left(v-6\right)v^{2}} have the same denominator, add them by adding their numerators.

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

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

$$\frac{3v^{3}+3v^{2}+6v-36}{3\left(v-6\right)v^{2}}$$

Factor the expressions that are not already factored in \frac{3v^{3}+3v^{2}+6v-36}{3\left(v-6\right)v^{2}}.

$$\frac{3\left(v^{3}+v^{2}+2v-12\right)}{3\left(v-6\right)v^{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{v^{3}+v^{2}+2v-12}{\left(v-6\right)v^{2}}$$

Expand \left(v-6\right)v^{2}.

$$\frac{v^{3}+v^{2}+2v-12}{v^{3}-6v^{2}}$$