Solve 2x^3+x^2-22x+24

$$2 x ^ { 3 } + x ^ { 2 } - 22 x + 24$$

Factor

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

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

$$\left(x+4\right)\left(2x^{2}-7x+6\right)$$

Consider 2x^{2}-7x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\times 6=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-4 b=-3$$

Rewrite 2x^{2}-7x+6 as \left(2x^{2}-4x\right)+\left(-3x+6\right).

$$\left(2x^{2}-4x\right)+\left(-3x+6\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(x-2\right)-3\left(x-2\right)$$

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

$$\left(x-2\right)\left(2x-3\right)$$

Rewrite the complete factored expression.

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

Evaluate

$$2x^{3}+x^{2}-22x+24$$