Solve t^5-t^3+t^2-1

$$t ^ { 5 } - t ^ { 3 } + t ^ { 2 } - 1$$

Factor

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

Solution Steps

Do the grouping t^{5}-t^{3}+t^{2}-1=\left(t^{5}-t^{3}\right)+\left(t^{2}-1\right), and factor out t^{3} in t^{5}-t^{3}.

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

Factor out common term t^{2}-1 by using distributive property.

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

Consider t^{2}-1. Rewrite t^{2}-1 as t^{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).

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

Consider t^{3}+1. Rewrite t^{3}+1 as t^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

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

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

Evaluate

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