Solve =x^{4}+x^3+x^2+x-x^3-x^2-x-1

$$= x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x - x ^ { 3 } - x ^ { 2 } - x - 1$$

Evaluate

$$x^{4}-1$$

Solution Steps

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

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

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

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

Combine x and -x to get 0.

$$x^{4}-1$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$x^{4}-1$$

Rewrite x^{4}-1 as \left(x^{2}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

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

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