Solve 2x^3+7x^2-7x-12

$$2 { x }^{ 3 } +7 { x }^{ 2 } -7x-12$$

Factor

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

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 -12 and q divides the leading coefficient 2. One such root is \frac{3}{2}. Factor the polynomial by dividing it by 2x-3.

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

Consider x^{2}+5x+4. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite x^{2}+5x+4 as \left(x^{2}+x\right)+\left(4x+4\right).

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

Factor out x in the first and 4 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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