Solve -8x^3-12x^2-6x-1

$$-8 { x }^{ 3 } -12 { x }^{ 2 } -6x-1$$

Factor

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

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 -1 and q divides the leading coefficient -8. One such root is -\frac{1}{2}. Factor the polynomial by dividing it by 2x+1.

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

Consider -4x^{2}-4x-1. Factor the expression by grouping. First, the expression needs to be rewritten as -4x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-4\left(-1\right)=4$$

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 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 -4.

$$a=-2 b=-2$$

Rewrite -4x^{2}-4x-1 as \left(-4x^{2}-2x\right)+\left(-2x-1\right).

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

Factor out -2x in the first and -1 in the second group.

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

Factor out common term 2x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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