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

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

Evaluate

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

Short Solution Steps

Factor x^{3}-1.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{-5x^{2}-2x}{x^{3}-1}$$

Factor

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