Solve f(x)=x^4-14x^3+34x^2+114x+65

$$f ( x ) = x ^ { 4 } - 14 x ^ { 3 } + 34 x ^ { 2 } + 114 x + 65$$

Factor

$$\left(x+1\right)^{2}\left(x^{2}-16x+65\right)$$

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

$$\left(x+1\right)\left(x^{3}-15x^{2}+49x+65\right)$$

Consider x^{3}-15x^{2}+49x+65. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 65 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{2}-16x+65\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-16x+65 is not factored since it does not have any rational roots.

$$\left(x^{2}-16x+65\right)\left(x+1\right)^{2}$$

Evaluate

$$\left(x+1\right)^{2}\left(x^{2}-16x+65\right)$$