Solve x^4-x^2-2x-1div(x^2+x+1)

$$x ^ { 4 } - x ^ { 2 } - 2 x - 1 \div ( x ^ { 2 } + x + 1 )$$

Evaluate

$$\frac{x^{6}+x^{5}-3x^{3}-3x^{2}-2x-1}{x^{2}+x+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4}-x^{2}-2x times \frac{x^{2}+x+1}{x^{2}+x+1}.

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

Since \frac{\left(x^{4}-x^{2}-2x\right)\left(x^{2}+x+1\right)}{x^{2}+x+1} and \frac{1}{x^{2}+x+1} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{x^{6}+x^{5}+x^{4}-x^{4}-x^{3}-x^{2}-2x^{3}-2x^{2}-2x-1}{x^{2}+x+1}$$

Combine like terms in x^{6}+x^{5}+x^{4}-x^{4}-x^{3}-x^{2}-2x^{3}-2x^{2}-2x-1.

$$\frac{-2x+x^{6}+x^{5}-3x^{3}-3x^{2}-1}{x^{2}+x+1}$$

Differentiate w.r.t. x

$$\frac{4x^{7}+8x^{6}+10x^{5}+2x^{4}-6x^{3}-10x^{2}-4x-1}{\left(x^{2}+x+1\right)^{2}}$$