Solve =x^4+6x^3+22x^2+30x+13

$$= x ^ { 4 } + 6 x ^ { 3 } + 22 x ^ { 2 } + 30 x + 13$$

Factor

$$\left(x+1\right)^{2}\left(x^{2}+4x+13\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 13 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{3}+5x^{2}+17x+13\right)$$

Consider x^{3}+5x^{2}+17x+13. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 13 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{2}+4x+13\right)$$

Rewrite the complete factored expression. Polynomial x^{2}+4x+13 is not factored since it does not have any rational roots.

$$\left(x^{2}+4x+13\right)\left(x+1\right)^{2}$$

Evaluate

$$\left(x+1\right)^{2}\left(x^{2}+4x+13\right)$$