Solve 9x^3-3x^2-5x-1

$$9 { x }^{ 3 } -3 { x }^{ 2 } -5x-1$$

Factor

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

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 9. One such root is -\frac{1}{3}. Factor the polynomial by dividing it by 3x+1.

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

Consider 3x^{2}-2x-1. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-2 ab=3\left(-1\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

Rewrite 3x^{2}-2x-1 as \left(3x^{2}-3x\right)+\left(x-1\right).

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

Factor out 3x in 3x^{2}-3x.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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