Solve -x^4-x^3+9x^2-11x+4

$$- x ^ { 4 } - x ^ { 3 } + 9 x ^ { 2 } - 11 x + 4$$

Factor

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

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

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

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

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

Consider -x^{2}+2x-1. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=2 ab=-\left(-1\right)=1$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=1$$

Rewrite -x^{2}+2x-1 as \left(-x^{2}+x\right)+\left(x-1\right).

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

Factor out -x in -x^{2}+x.

$$-x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(-x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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