Solve p/3p-4-2/pp-6

$$\frac { p } { 3 p - 4 } - \frac { 2 } { p p - 6 }$$

Evaluate

$$\frac{p^{3}-12p+8}{\left(3p-4\right)\left(p^{2}-6\right)}$$

Short Solution Steps

Multiply p and p to get p^{2}.

$$\frac{p}{3p-4}-\frac{2}{p^{2}-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3p-4 and p^{2}-6 is \left(3p-4\right)\left(p^{2}-6\right). Multiply \frac{p}{3p-4} times \frac{p^{2}-6}{p^{2}-6}. Multiply \frac{2}{p^{2}-6} times \frac{3p-4}{3p-4}.

$$\frac{p\left(p^{2}-6\right)}{\left(3p-4\right)\left(p^{2}-6\right)}-\frac{2\left(3p-4\right)}{\left(3p-4\right)\left(p^{2}-6\right)}$$

Since \frac{p\left(p^{2}-6\right)}{\left(3p-4\right)\left(p^{2}-6\right)} and \frac{2\left(3p-4\right)}{\left(3p-4\right)\left(p^{2}-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{p\left(p^{2}-6\right)-2\left(3p-4\right)}{\left(3p-4\right)\left(p^{2}-6\right)}$$

Do the multiplications in p\left(p^{2}-6\right)-2\left(3p-4\right).

$$\frac{p^{3}-6p-6p+8}{\left(3p-4\right)\left(p^{2}-6\right)}$$

Combine like terms in p^{3}-6p-6p+8.

$$\frac{p^{3}-12p+8}{\left(3p-4\right)\left(p^{2}-6\right)}$$

Expand \left(3p-4\right)\left(p^{2}-6\right).

$$\frac{p^{3}-12p+8}{3p^{3}-4p^{2}-18p+24}$$

Differentiate w.r.t. p

$$\frac{4\left(-p^{4}+9p^{3}-12p^{2}+16p-36\right)}{\left(\left(3p-4\right)\left(p^{2}-6\right)\right)^{2}}$$