Solve x^3+4x^2+x-6/x^3-7x+6-2x+5x-3/2x^2-2x

$$\frac { x ^ { 3 } + 4 x ^ { 2 } + x - 6 } { x ^ { 3 } - 7 x + 6 } - \frac { 2 x + 5 x - 3 } { 2 x ^ { 2 } - 2 x }$$

Evaluate

$$\frac{2x^{3}-5x^{2}+13x-6}{2x\left(x-2\right)\left(x-1\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}+4x^{2}+x-6}{x^{3}-7x+6}.

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

Cancel out \left(x-1\right)\left(x+3\right) in both numerator and denominator.

$$\frac{x+2}{x-2}-\frac{2x+5x-3}{2x^{2}-2x}$$

Combine 2x and 5x to get 7x.

$$\frac{x+2}{x-2}-\frac{7x-3}{2x^{2}-2x}$$

Factor 2x^{2}-2x.

$$\frac{x+2}{x-2}-\frac{7x-3}{2x\left(x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and 2x\left(x-1\right) is 2x\left(x-2\right)\left(x-1\right). Multiply \frac{x+2}{x-2} times \frac{2x\left(x-1\right)}{2x\left(x-1\right)}. Multiply \frac{7x-3}{2x\left(x-1\right)} times \frac{x-2}{x-2}.

$$\frac{\left(x+2\right)\times 2x\left(x-1\right)}{2x\left(x-2\right)\left(x-1\right)}-\frac{\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(x+2\right)\times 2x\left(x-1\right)}{2x\left(x-2\right)\left(x-1\right)} and \frac{\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+2\right)\times 2x\left(x-1\right)-\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(x+2\right)\times 2x\left(x-1\right)-\left(7x-3\right)\left(x-2\right).

$$\frac{2x^{3}-2x^{2}+4x^{2}-4x-7x^{2}+14x+3x-6}{2x\left(x-2\right)\left(x-1\right)}$$

Combine like terms in 2x^{3}-2x^{2}+4x^{2}-4x-7x^{2}+14x+3x-6.

$$\frac{2x^{3}-5x^{2}+13x-6}{2x\left(x-2\right)\left(x-1\right)}$$

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

$$\frac{2x^{3}-5x^{2}+13x-6}{2x^{3}-6x^{2}+4x}$$

Expand

$$\frac{2x^{3}-5x^{2}+13x-6}{2\left(x-2\right)\left(x^{2}-x\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}+4x^{2}+x-6}{x^{3}-7x+6}.

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

Cancel out \left(x-1\right)\left(x+3\right) in both numerator and denominator.

$$\frac{x+2}{x-2}-\frac{2x+5x-3}{2x^{2}-2x}$$

Combine 2x and 5x to get 7x.

$$\frac{x+2}{x-2}-\frac{7x-3}{2x^{2}-2x}$$

Factor 2x^{2}-2x.

$$\frac{x+2}{x-2}-\frac{7x-3}{2x\left(x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and 2x\left(x-1\right) is 2x\left(x-2\right)\left(x-1\right). Multiply \frac{x+2}{x-2} times \frac{2x\left(x-1\right)}{2x\left(x-1\right)}. Multiply \frac{7x-3}{2x\left(x-1\right)} times \frac{x-2}{x-2}.

$$\frac{\left(x+2\right)\times 2x\left(x-1\right)}{2x\left(x-2\right)\left(x-1\right)}-\frac{\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(x+2\right)\times 2x\left(x-1\right)}{2x\left(x-2\right)\left(x-1\right)} and \frac{\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+2\right)\times 2x\left(x-1\right)-\left(7x-3\right)\left(x-2\right)}{2x\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(x+2\right)\times 2x\left(x-1\right)-\left(7x-3\right)\left(x-2\right).

$$\frac{2x^{3}-2x^{2}+4x^{2}-4x-7x^{2}+14x+3x-6}{2x\left(x-2\right)\left(x-1\right)}$$

Combine like terms in 2x^{3}-2x^{2}+4x^{2}-4x-7x^{2}+14x+3x-6.

$$\frac{2x^{3}-5x^{2}+13x-6}{2x\left(x-2\right)\left(x-1\right)}$$

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

$$\frac{2x^{3}-5x^{2}+13x-6}{2x^{3}-6x^{2}+4x}$$