Solve F(s)=6/s-1/s-8+4/s-3

$$F ( s ) = \frac { 6 } { s } - \frac { 1 } { s - 8 } + \frac { 4 } { s - 3 }$$

Evaluate

$$\frac{9s^{2}-95s+144}{s\left(s-8\right)\left(s-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s and s-8 is s\left(s-8\right). Multiply \frac{6}{s} times \frac{s-8}{s-8}. Multiply \frac{1}{s-8} times \frac{s}{s}.

$$\frac{6\left(s-8\right)}{s\left(s-8\right)}-\frac{s}{s\left(s-8\right)}+\frac{4}{s-3}$$

Since \frac{6\left(s-8\right)}{s\left(s-8\right)} and \frac{s}{s\left(s-8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\left(s-8\right)-s}{s\left(s-8\right)}+\frac{4}{s-3}$$

Do the multiplications in 6\left(s-8\right)-s.

$$\frac{6s-48-s}{s\left(s-8\right)}+\frac{4}{s-3}$$

Combine like terms in 6s-48-s.

$$\frac{5s-48}{s\left(s-8\right)}+\frac{4}{s-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s\left(s-8\right) and s-3 is s\left(s-8\right)\left(s-3\right). Multiply \frac{5s-48}{s\left(s-8\right)} times \frac{s-3}{s-3}. Multiply \frac{4}{s-3} times \frac{s\left(s-8\right)}{s\left(s-8\right)}.

$$\frac{\left(5s-48\right)\left(s-3\right)}{s\left(s-8\right)\left(s-3\right)}+\frac{4s\left(s-8\right)}{s\left(s-8\right)\left(s-3\right)}$$

Since \frac{\left(5s-48\right)\left(s-3\right)}{s\left(s-8\right)\left(s-3\right)} and \frac{4s\left(s-8\right)}{s\left(s-8\right)\left(s-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(5s-48\right)\left(s-3\right)+4s\left(s-8\right)}{s\left(s-8\right)\left(s-3\right)}$$

Do the multiplications in \left(5s-48\right)\left(s-3\right)+4s\left(s-8\right).

$$\frac{5s^{2}-15s-48s+144+4s^{2}-32s}{s\left(s-8\right)\left(s-3\right)}$$

Combine like terms in 5s^{2}-15s-48s+144+4s^{2}-32s.

$$\frac{9s^{2}-95s+144}{s\left(s-8\right)\left(s-3\right)}$$

Expand s\left(s-8\right)\left(s-3\right).

$$\frac{9s^{2}-95s+144}{s^{3}-11s^{2}+24s}$$

Differentiate w.r.t. s

$$\frac{-9s^{4}+190s^{3}-1261s^{2}+3168s-3456}{\left(s\left(s-8\right)\left(s-3\right)\right)^{2}}$$