Solve k^4+k^3+k+1=

$$k ^ { 4 } + k ^ { 3 } + k + 1 =$$

Factor

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

Solution Steps

Do the grouping k^{4}+k^{3}+k+1=\left(k^{4}+k^{3}\right)+\left(k+1\right), and factor out k^{3} in k^{4}+k^{3}.

$$k^{3}\left(k+1\right)+k+1$$

Factor out common term k+1 by using distributive property.

$$\left(k+1\right)\left(k^{3}+1\right)$$

Consider k^{3}+1. Rewrite k^{3}+1 as k^{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).

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

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

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

Evaluate

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