Solve P(x)=x^4+9x^3+23x^2+8x-16

$$P ( x ) = x ^ { 4 } + 9 x ^ { 3 } + 23 x ^ { 2 } + 8 x - 16$$

Factor

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

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 -16 and q divides the leading coefficient 1. One such root is -4. Factor the polynomial by dividing it by x+4.

$$\left(x+4\right)\left(x^{3}+5x^{2}+3x-4\right)$$

Consider x^{3}+5x^{2}+3x-4. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. One such root is -4. Factor the polynomial by dividing it by x+4.

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

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

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

Evaluate

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