Solve f(x)=-2x^4+13x^3-21x^2+2x+8

$$f ( x ) = - 2 x ^ { 4 } + 13 x ^ { 3 } - 21 x ^ { 2 } + 2 x + 8$$

Factor

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

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 8 and q divides the leading coefficient -2. One such root is 4. Factor the polynomial by dividing it by x-4.

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

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

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

Consider -2x^{2}+3x+2. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=3 ab=-2\times 2=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$a=4 b=-1$$

Rewrite -2x^{2}+3x+2 as \left(-2x^{2}+4x\right)+\left(-x+2\right).

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

Factor out 2x in -2x^{2}+4x.

$$2x\left(-x+2\right)-x+2$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(2x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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