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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 1-x is x\left(-x+1\right). Multiply \frac{1-x^{2}}{x} times \frac{-x+1}{-x+1}. Multiply \frac{6}{1-x} times \frac{x}{x}.

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

Since \frac{\left(1-x^{2}\right)\left(-x+1\right)}{x\left(-x+1\right)} and \frac{6x}{x\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in 1-x+x^{3}-x^{2}-6x.

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

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

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

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

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

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

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

Combine like terms in -x+7x^{2}-x^{4}+x^{3}-x^{2}+x+1-x.

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

Expand \left(x-1\right)x^{2}.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 1-x is x\left(-x+1\right). Multiply \frac{1-x^{2}}{x} times \frac{-x+1}{-x+1}. Multiply \frac{6}{1-x} times \frac{x}{x}.

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

Since \frac{\left(1-x^{2}\right)\left(-x+1\right)}{x\left(-x+1\right)} and \frac{6x}{x\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in 1-x+x^{3}-x^{2}-6x.

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

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

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

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

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

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

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

Combine like terms in -x+7x^{2}-x^{4}+x^{3}-x^{2}+x+1-x.

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

Expand \left(x-1\right)x^{2}.

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