Solve 2x^4-18x^2-8x+24

$$2 { x }^{ 4 } -18 { x }^{ 2 } -8x+24$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(x^{4}-9x^{2}-4x+12\right)$$

Consider x^{4}-9x^{2}-4x+12. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 12 and q divides the leading coefficient 1. One such root is 3. Factor the polynomial by dividing it by x-3.

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

Consider x^{3}+3x^{2}-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 -2. Factor the polynomial by dividing it by x+2.

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

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

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

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. The only such pair is the system solution.

$$a=-1 b=2$$

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

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

Factor out x in the first and 2 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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