Solve 1/x-1+2/x-2+3/x-3+4/x-4

$$\frac{ 1 }{ x-1 } + \frac{ 2 }{ x-2 } + \frac{ 3 }{ x-3 } + \frac{ 4 }{ x-4 }$$

Evaluate

$$\frac{2\left(5x^{3}-35x^{2}+75x-48\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}$$

Short Solution Steps

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

$$\frac{x-2}{\left(x-2\right)\left(x-1\right)}+\frac{2\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}+\frac{3}{x-3}+\frac{4}{x-4}$$

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

$$\frac{x-2+2\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}+\frac{3}{x-3}+\frac{4}{x-4}$$

Do the multiplications in x-2+2\left(x-1\right).

$$\frac{x-2+2x-2}{\left(x-2\right)\left(x-1\right)}+\frac{3}{x-3}+\frac{4}{x-4}$$

Combine like terms in x-2+2x-2.

$$\frac{3x-4}{\left(x-2\right)\left(x-1\right)}+\frac{3}{x-3}+\frac{4}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right) and x-3 is \left(x-3\right)\left(x-2\right)\left(x-1\right). Multiply \frac{3x-4}{\left(x-2\right)\left(x-1\right)} times \frac{x-3}{x-3}. Multiply \frac{3}{x-3} times \frac{\left(x-2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(3x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{3\left(x-2\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{4}{x-4}$$

Since \frac{\left(3x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} and \frac{3\left(x-2\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x-4\right)\left(x-3\right)+3\left(x-2\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{4}{x-4}$$

Do the multiplications in \left(3x-4\right)\left(x-3\right)+3\left(x-2\right)\left(x-1\right).

$$\frac{3x^{2}-9x-4x+12+3x^{2}-3x-6x+6}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{4}{x-4}$$

Combine like terms in 3x^{2}-9x-4x+12+3x^{2}-3x-6x+6.

$$\frac{6x^{2}-22x+18}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{4}{x-4}$$

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)\left(x-1\right) and x-4 is \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right). Multiply \frac{6x^{2}-22x+18}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} times \frac{x-4}{x-4}. Multiply \frac{4}{x-4} times \frac{\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(6x^{2}-22x+18\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{4\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(6x^{2}-22x+18\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)} and \frac{4\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(6x^{2}-22x+18\right)\left(x-4\right)+4\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(6x^{2}-22x+18\right)\left(x-4\right)+4\left(x-3\right)\left(x-2\right)\left(x-1\right).

$$\frac{6x^{3}-24x^{2}-22x^{2}+88x+18x-72+4x^{3}-12x^{2}+8x-12x^{2}+36x-24}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}$$

Combine like terms in 6x^{3}-24x^{2}-22x^{2}+88x+18x-72+4x^{3}-12x^{2}+8x-12x^{2}+36x-24.

$$\frac{10x^{3}-70x^{2}+150x-96}{\left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right)}$$

Expand \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right).

$$\frac{10x^{3}-70x^{2}+150x-96}{x^{4}-10x^{3}+35x^{2}-50x+24}$$