Solve 6x^6/x^2-4x+3-3/3-x-4/x-1

$$\frac { 6 x ^ { 6 } } { x ^ { 2 } - 4 x + 3 } - \frac { 3 } { 3 - x } - \frac { 4 } { x - 1 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-4x+3.

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

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

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

Since \frac{6x^{6}}{\left(x-3\right)\left(x-1\right)} and \frac{3\left(-1\right)\left(x-1\right)}{\left(x-3\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 6x^{6}-3\left(-1\right)\left(x-1\right).

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

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

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

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

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

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

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

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

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

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

$$\frac{-x+6x^{6}+9}{x^{2}-4x+3}$$

Differentiate w.r.t. x

$$\frac{24x^{7}-120x^{6}+108x^{5}+x^{2}-18x+33}{\left(\left(x-3\right)\left(x-1\right)\right)^{2}}$$