Solve frac{x^{5}-1}{3x^{5}+3x^4+3x^3+3x^2+3x}:6x+6/4x^2-4

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

Evaluate

$$\frac{2\left(x-1\right)^{2}}{9x}$$

Short Solution Steps

Divide \frac{x^{5}-1}{3x^{5}+3x^{4}+3x^{3}+3x^{2}+3x} by \frac{6x+6}{4x^{2}-4} by multiplying \frac{x^{5}-1}{3x^{5}+3x^{4}+3x^{3}+3x^{2}+3x} by the reciprocal of \frac{6x+6}{4x^{2}-4}.

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

Factor the expressions that are not already factored.

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

Cancel out 2\left(x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right) in both numerator and denominator.

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

Expand the expression.

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

Expand

$$\frac{2x}{9}-\frac{4}{9}+\frac{2}{9x}$$

Short Solution Steps

Divide \frac{x^{5}-1}{3x^{5}+3x^{4}+3x^{3}+3x^{2}+3x} by \frac{6x+6}{4x^{2}-4} by multiplying \frac{x^{5}-1}{3x^{5}+3x^{4}+3x^{3}+3x^{2}+3x} by the reciprocal of \frac{6x+6}{4x^{2}-4}.

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

Factor the expressions that are not already factored.

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

Cancel out 2\left(x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right) in both numerator and denominator.

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

Expand the expression.

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