Solve frac{x^{4}}{x}+frac{(1-x^{4})(x-x^{4})}{x-x^{3}}+frac{(1-x^{4})(x-x^{4})(x^{2}-x^{4})}{x^{3}-x^{6}}+frac{(1-x^{4})(x-x^{4})(x^{2}-x^{4})(x^3-x^4)}{x^6-x^10}

$$\frac { x ^ { 4 } } { x } + \frac { ( 1 - x ^ { 4 } ) ( x - x ^ { 4 } ) } { x - x ^ { 3 } } + \frac { ( 1 - x ^ { 4 } ) ( x - x ^ { 4 } ) ( x ^ { 2 } - x ^ { 4 } ) } { x ^ { 3 } - x ^ { 6 } } + \frac { ( 1 - x ^ { 4 } ) ( x - x ^ { 4 } ) ( x ^ { 2 } - x ^ { 4 } ) ( x ^ { 3 } - x ^ { 4 } ) } { x ^ { 6 } - x ^ { 10 } }$$

Evaluate

$$3-x-x^{2}$$

Short Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 4 to get 3.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)}{x-x^{3}}.

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

Extract the negative sign in 1+x.

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

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

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

Expand the expression.

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

Combine x^{3} and -x^{3} to get 0.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)\left(x^{2}-x^{4}\right)}{x^{3}-x^{6}}.

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

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

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

Expand the expression.

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

Combine x^{2} and -x^{2} to get 0.

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

Add 1 and 1 to get 2.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)\left(x^{2}-x^{4}\right)\left(x^{3}-x^{4}\right)}{x^{6}-x^{10}}.

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

Extract the negative sign in -1-x. Extract the negative sign in 1-x.

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

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

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

Expand the expression.

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

Combine x^{6} and -x^{6} to get 0.

$$-x^{5}+2-x^{4}+x^{5}+x^{4}-x^{2}-x+1$$

Combine -x^{5} and x^{5} to get 0.

$$2-x^{4}+x^{4}-x^{2}-x+1$$

Combine -x^{4} and x^{4} to get 0.

$$2-x^{2}-x+1$$

Add 2 and 1 to get 3.

$$3-x^{2}-x$$

Expand

$$3-x-x^{2}$$

Short Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 4 to get 3.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)}{x-x^{3}}.

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

Extract the negative sign in 1+x.

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

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

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

Expand the expression.

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

Combine x^{3} and -x^{3} to get 0.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)\left(x^{2}-x^{4}\right)}{x^{3}-x^{6}}.

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

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

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

Expand the expression.

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

Combine x^{2} and -x^{2} to get 0.

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

Add 1 and 1 to get 2.

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

Factor the expressions that are not already factored in \frac{\left(1-x^{4}\right)\left(x-x^{4}\right)\left(x^{2}-x^{4}\right)\left(x^{3}-x^{4}\right)}{x^{6}-x^{10}}.

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

Extract the negative sign in -1-x. Extract the negative sign in 1-x.

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

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

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

Expand the expression.

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

Combine x^{6} and -x^{6} to get 0.

$$-x^{5}+2-x^{4}+x^{5}+x^{4}-x^{2}-x+1$$

Combine -x^{5} and x^{5} to get 0.

$$2-x^{4}+x^{4}-x^{2}-x+1$$

Combine -x^{4} and x^{4} to get 0.

$$2-x^{2}-x+1$$

Add 2 and 1 to get 3.

$$3-x^{2}-x$$