Solve 5x^3+27x^2-20x-12

$$5 x ^ { 3 } + 27 x ^ { 2 } - 20 x - 12$$

Factor

$$\left(x-1\right)\left(x+6\right)\left(5x+2\right)$$

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 5. One such root is -\frac{2}{5}. Factor the polynomial by dividing it by 5x+2.

$$\left(5x+2\right)\left(x^{2}+5x-6\right)$$

Consider x^{2}+5x-6. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-6\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=-1 b=6$$

Rewrite x^{2}+5x-6 as \left(x^{2}-x\right)+\left(6x-6\right).

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

Factor out x in the first and 6 in the second group.

$$x\left(x-1\right)+6\left(x-1\right)$$

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

$$\left(x-1\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$\left(x-1\right)\left(5x+2\right)\left(x+6\right)$$

Evaluate

$$\left(x-1\right)\left(x+6\right)\left(5x+2\right)$$