Solve x^4-x^3-x^3+x

$$x ^ { 4 } - x ^ { 3 } - x ^ { 3 } + x$$

Factor

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

Solution Steps

Factor out x.

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

Consider x^{3}-x^{2}-x^{2}+1. Multiply and combine like terms.

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

Consider x^{3}-2x^{2}+1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by x-1.

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

Rewrite the complete factored expression. Polynomial x^{2}-x-1 is not factored since it does not have any rational roots.

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

Evaluate

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

Solution Steps

Combine -x^{3} and -x^{3} to get -2x^{3}.

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