Solve f(x)=x^5+3x^4+5x^3+15x^2+4x+12

$$f ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } + 4 x + 12$$

Factor

$$\left(x+3\right)\left(x^{2}+1\right)\left(x^{2}+4\right)$$

Solution Steps

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{5} and m divides the constant factor 12. One such factor is x^{2}+4. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}+4\right)\left(x^{3}+3x^{2}+x+3\right)$$

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. The following polynomials are not factored since they do not have any rational roots: x^{2}+1,x^{2}+4.

$$\left(x^{2}+1\right)\left(x+3\right)\left(x^{2}+4\right)$$

Evaluate

$$\left(x+3\right)\left(x^{2}+1\right)\left(x^{2}+4\right)$$