Solve 8z+2/3z^2-3z-18+9/3z^2-12z+9

$$\frac { 8 z + 2 } { 3 z ^ { 2 } - 3 z - 18 } + \frac { 9 } { 3 z ^ { 2 } - 12 z + 9 }$$

Evaluate

$$\frac{8z^{2}+3z+16}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Short Solution Steps

Factor 3z^{2}-3z-18. Factor 3z^{2}-12z+9.

$$\frac{8z+2}{3\left(z-3\right)\left(z+2\right)}+\frac{9}{3\left(z-3\right)\left(z-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(z-3\right)\left(z+2\right) and 3\left(z-3\right)\left(z-1\right) is 3\left(z-3\right)\left(z-1\right)\left(z+2\right). Multiply \frac{8z+2}{3\left(z-3\right)\left(z+2\right)} times \frac{z-1}{z-1}. Multiply \frac{9}{3\left(z-3\right)\left(z-1\right)} times \frac{z+2}{z+2}.

$$\frac{\left(8z+2\right)\left(z-1\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}+\frac{9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Since \frac{\left(8z+2\right)\left(z-1\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)} and \frac{9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(8z+2\right)\left(z-1\right)+9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Do the multiplications in \left(8z+2\right)\left(z-1\right)+9\left(z+2\right).

$$\frac{8z^{2}-8z+2z-2+9z+18}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Combine like terms in 8z^{2}-8z+2z-2+9z+18.

$$\frac{8z^{2}+3z+16}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Expand 3\left(z-3\right)\left(z-1\right)\left(z+2\right).

$$\frac{8z^{2}+3z+16}{3z^{3}-6z^{2}-15z+18}$$

Expand

$$\frac{8z^{2}+3z+16}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Short Solution Steps

Factor 3z^{2}-3z-18. Factor 3z^{2}-12z+9.

$$\frac{8z+2}{3\left(z-3\right)\left(z+2\right)}+\frac{9}{3\left(z-3\right)\left(z-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(z-3\right)\left(z+2\right) and 3\left(z-3\right)\left(z-1\right) is 3\left(z-3\right)\left(z-1\right)\left(z+2\right). Multiply \frac{8z+2}{3\left(z-3\right)\left(z+2\right)} times \frac{z-1}{z-1}. Multiply \frac{9}{3\left(z-3\right)\left(z-1\right)} times \frac{z+2}{z+2}.

$$\frac{\left(8z+2\right)\left(z-1\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}+\frac{9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Since \frac{\left(8z+2\right)\left(z-1\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)} and \frac{9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(8z+2\right)\left(z-1\right)+9\left(z+2\right)}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Do the multiplications in \left(8z+2\right)\left(z-1\right)+9\left(z+2\right).

$$\frac{8z^{2}-8z+2z-2+9z+18}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Combine like terms in 8z^{2}-8z+2z-2+9z+18.

$$\frac{8z^{2}+3z+16}{3\left(z-3\right)\left(z-1\right)\left(z+2\right)}$$

Expand 3\left(z-3\right)\left(z-1\right)\left(z+2\right).

$$\frac{8z^{2}+3z+16}{3z^{3}-6z^{2}-15z+18}$$