Solve 12x^2-5x^3-x-6

$$12 x ^ { 2 } - 5 x ^ { 3 } - x - 6$$

Factor

$$\left(1-x\right)\left(x-2\right)\left(5x+3\right)$$

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 -6 and q divides the leading coefficient -5. One such root is -\frac{3}{5}. Factor the polynomial by dividing it by 5x+3.

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

Consider -x^{2}+3x-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.

$$a+b=3 ab=-\left(-2\right)=2$$

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.

$$a=2 b=1$$

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

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

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

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

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

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

Rewrite the complete factored expression.

$$\left(x-2\right)\left(-x+1\right)\left(5x+3\right)$$

Evaluate

$$\left(1-x\right)\left(x-2\right)\left(5x+3\right)$$