Solve x-4/x^2-5x+6+x+7/x^2-16

$$\frac { x - 4 } { x ^ { 2 } - 5 x + 6 } + \frac { x + 7 } { x ^ { 2 } - 16 }$$

Evaluate

$$\frac{2x^{3}-2x^{2}-45x+106}{\left(x-3\right)\left(x-2\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}-5x+6. Factor x^{2}-16.

$$\frac{x-4}{\left(x-3\right)\left(x-2\right)}+\frac{x+7}{\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x-2\right) and \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right). Multiply \frac{x-4}{\left(x-3\right)\left(x-2\right)} times \frac{\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}. Multiply \frac{x+7}{\left(x-4\right)\left(x+4\right)} times \frac{\left(x-3\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}.

$$\frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}+\frac{\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Since \frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)} and \frac{\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)+\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Do the multiplications in \left(x-4\right)\left(x-4\right)\left(x+4\right)+\left(x+7\right)\left(x-3\right)\left(x-2\right).

$$\frac{x^{3}-16x-4x^{2}+64+x^{3}-5x^{2}+6x+7x^{2}-35x+42}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Combine like terms in x^{3}-16x-4x^{2}+64+x^{3}-5x^{2}+6x+7x^{2}-35x+42.

$$\frac{2x^{3}-45x-2x^{2}+106}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right).

$$\frac{2x^{3}-45x-2x^{2}+106}{x^{4}-5x^{3}-10x^{2}+80x-96}$$

Expand

$$\frac{2x^{3}-2x^{2}-45x+106}{\left(x^{2}-16\right)\left(x^{2}-5x+6\right)}$$

Short Solution Steps

Factor x^{2}-5x+6. Factor x^{2}-16.

$$\frac{x-4}{\left(x-3\right)\left(x-2\right)}+\frac{x+7}{\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x-2\right) and \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right). Multiply \frac{x-4}{\left(x-3\right)\left(x-2\right)} times \frac{\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}. Multiply \frac{x+7}{\left(x-4\right)\left(x+4\right)} times \frac{\left(x-3\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}.

$$\frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}+\frac{\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Since \frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)} and \frac{\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-4\right)\left(x-4\right)\left(x+4\right)+\left(x+7\right)\left(x-3\right)\left(x-2\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Do the multiplications in \left(x-4\right)\left(x-4\right)\left(x+4\right)+\left(x+7\right)\left(x-3\right)\left(x-2\right).

$$\frac{x^{3}-16x-4x^{2}+64+x^{3}-5x^{2}+6x+7x^{2}-35x+42}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Combine like terms in x^{3}-16x-4x^{2}+64+x^{3}-5x^{2}+6x+7x^{2}-35x+42.

$$\frac{2x^{3}-45x-2x^{2}+106}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x+4\right).

$$\frac{2x^{3}-45x-2x^{2}+106}{x^{4}-5x^{3}-10x^{2}+80x-96}$$