Solve P(x)=x^5+3x^4+x^3+3x^2

$$P ( x ) = x ^ { 5 } + 3 x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out x^{2}.

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

Consider x^{3}+3x^{2}+x+3. Do the grouping x^{3}+3x^{2}+x+3=\left(x^{3}+3x^{2}\right)+\left(x+3\right), and factor out x^{2} in x^{3}+3x^{2}.

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

Factor out common term x+3 by using distributive property.

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

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

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

Evaluate

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