Solve h(x)=x^4-18x^3+78x^2+294x-3355

$$h ( x ) = x ^ { 4 } - 18 x ^ { 3 } + 78 x ^ { 2 } + 294 x - 3355$$

Factor

$$\left(x-11\right)\left(x+5\right)\left(x^{2}-12x+61\right)$$

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 -3355 and q divides the leading coefficient 1. One such root is 11. Factor the polynomial by dividing it by x-11.

$$\left(x-11\right)\left(x^{3}-7x^{2}+x+305\right)$$

Consider x^{3}-7x^{2}+x+305. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 305 and q divides the leading coefficient 1. One such root is -5. Factor the polynomial by dividing it by x+5.

$$\left(x+5\right)\left(x^{2}-12x+61\right)$$

Rewrite the complete factored expression. Polynomial x^{2}-12x+61 is not factored since it does not have any rational roots.

$$\left(x-11\right)\left(x+5\right)\left(x^{2}-12x+61\right)$$

Evaluate

$$\left(x-11\right)\left(x+5\right)\left(x^{2}-12x+61\right)$$