Solve 95/(x+2)+93/x^24-4

$$\frac { 95 } { ( x + 2 ) } + \frac { 93 } { x ^ { 24 } - 4 }$$

Evaluate

$$\frac{95x^{24}+93x-194}{\left(x+2\right)\left(x^{24}-4\right)}$$

Short Solution Steps

Factor x^{24}-4.

$$\frac{95}{x+2}+\frac{93}{\left(x^{12}-2\right)\left(x^{12}+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and \left(x^{12}-2\right)\left(x^{12}+2\right) is \left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right). Multiply \frac{95}{x+2} times \frac{\left(x^{12}-2\right)\left(x^{12}+2\right)}{\left(x^{12}-2\right)\left(x^{12}+2\right)}. Multiply \frac{93}{\left(x^{12}-2\right)\left(x^{12}+2\right)} times \frac{x+2}{x+2}.

$$\frac{95\left(x^{12}-2\right)\left(x^{12}+2\right)}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)}+\frac{93\left(x+2\right)}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)}$$

Since \frac{95\left(x^{12}-2\right)\left(x^{12}+2\right)}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)} and \frac{93\left(x+2\right)}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{95\left(x^{12}-2\right)\left(x^{12}+2\right)+93\left(x+2\right)}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)}$$

Do the multiplications in 95\left(x^{12}-2\right)\left(x^{12}+2\right)+93\left(x+2\right).

$$\frac{95x^{24}+190x^{12}-190x^{12}-380+93x+186}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)}$$

Combine like terms in 95x^{24}+190x^{12}-190x^{12}-380+93x+186.

$$\frac{-194+95x^{24}+93x}{\left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right)}$$

Expand \left(x+2\right)\left(x^{12}-2\right)\left(x^{12}+2\right).

$$\frac{-194+95x^{24}+93x}{x^{25}+2x^{24}-4x-8}$$

Differentiate w.r.t. x

$$-\frac{95x^{48}+2232x^{25}+8168x^{24}+8928x^{23}+1520}{\left(\left(x+2\right)\left(x^{24}-4\right)\right)^{2}}$$