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

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

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

Extract the negative sign in 1+x.

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

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

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

Expand the expression.

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

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}}.

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

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

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

Expand the expression.

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

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

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

Add 1 and 1 to get 2.

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

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}}.

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

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

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

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.

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

Expand the expression.

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

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

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

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

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

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

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

Add 2 and 1 to get 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{3}+3-x^{2}-x times \frac{-x}{-x}.

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

Since \frac{-x^{4}}{-x} and \frac{\left(-x^{3}+3-x^{2}-x\right)\left(-1\right)x}{-x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x^{4}+\left(-x^{3}+3-x^{2}-x\right)\left(-1\right)x.

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

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

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

Factor the expressions that are not already factored in \frac{-3x+x^{3}+x^{2}}{-x}.

$$\frac{x\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)}{-x}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)$$

To find the opposite of -\frac{1}{2}\sqrt{13}-\frac{1}{2}, find the opposite of each term.

$$-\left(x+\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)$$

To find the opposite of \frac{1}{2}\sqrt{13}-\frac{1}{2}, find the opposite of each term.

$$-\left(x+\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)$$

Use the distributive property to multiply -1 by x+\frac{1}{2}\sqrt{13}+\frac{1}{2}.

$$\left(-x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)$$

Use the distributive property to multiply -x-\frac{1}{2}\sqrt{13}-\frac{1}{2} by x-\frac{1}{2}\sqrt{13}+\frac{1}{2} and combine like terms.

$$-x^{2}-x+\frac{1}{4}\left(\sqrt{13}\right)^{2}-\frac{1}{4}$$

The square of \sqrt{13} is 13.

$$-x^{2}-x+\frac{1}{4}\times 13-\frac{1}{4}$$

Multiply \frac{1}{4} and 13 to get \frac{13}{4}.

$$-x^{2}-x+\frac{13}{4}-\frac{1}{4}$$

Subtract \frac{1}{4} from \frac{13}{4} to get 3.

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

Expand

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

Short Solution Steps

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

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

Extract the negative sign in 1+x.

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

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

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

Expand the expression.

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

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}}.

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

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

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

Expand the expression.

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

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

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

Add 1 and 1 to get 2.

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

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}}.

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

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

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

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.

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

Expand the expression.

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

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

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

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

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

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

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

Add 2 and 1 to get 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{3}+3-x^{2}-x times \frac{-x}{-x}.

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

Since \frac{-x^{4}}{-x} and \frac{\left(-x^{3}+3-x^{2}-x\right)\left(-1\right)x}{-x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x^{4}+\left(-x^{3}+3-x^{2}-x\right)\left(-1\right)x.

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

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

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

Factor the expressions that are not already factored in \frac{-3x+x^{3}+x^{2}}{-x}.

$$\frac{x\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)}{-x}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(x-\left(-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)$$

To find the opposite of -\frac{1}{2}\sqrt{13}-\frac{1}{2}, find the opposite of each term.

$$-\left(x+\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\right)$$

To find the opposite of \frac{1}{2}\sqrt{13}-\frac{1}{2}, find the opposite of each term.

$$-\left(x+\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)$$

Use the distributive property to multiply -1 by x+\frac{1}{2}\sqrt{13}+\frac{1}{2}.

$$\left(-x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)$$

Use the distributive property to multiply -x-\frac{1}{2}\sqrt{13}-\frac{1}{2} by x-\frac{1}{2}\sqrt{13}+\frac{1}{2} and combine like terms.

$$-x^{2}-x+\frac{1}{4}\left(\sqrt{13}\right)^{2}-\frac{1}{4}$$

The square of \sqrt{13} is 13.

$$-x^{2}-x+\frac{1}{4}\times 13-\frac{1}{4}$$

Multiply \frac{1}{4} and 13 to get \frac{13}{4}.

$$-x^{2}-x+\frac{13}{4}-\frac{1}{4}$$

Subtract \frac{1}{4} from \frac{13}{4} to get 3.

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