Question

which product of prime polynomials is equivalent to ( 36x^3 - 15x^2 - 6x )?
( 3(x^2 + 1)(4x - 1) )
( 3x(3x - 2)(4x + 1) )
( x(3x - 2)(4x + 1) )
( 3(x^2 + 1)(4x + 1) )

Explanation:

Step1: Factor out GCF

First, find the greatest common factor (GCF) of $36x^3$, $-15x^2$, and $-6x$. The GCF is $3x$.
$36x^3 - 15x^2 - 6x = 3x(12x^2 - 5x - 2)$

Step2: Factor quadratic trinomial

Factor $12x^2 - 5x - 2$ by finding two numbers that multiply to $12\times(-2)=-24$ and add to $-5$. These numbers are $-8$ and $3$. Split the middle term and factor:
$12x^2 -8x +3x -2 = 4x(3x-2)+1(3x-2)=(3x-2)(4x+1)$

Step3: Combine factors

Substitute the factored quadratic back into the expression from Step1.
$3x(12x^2 - 5x - 2)=3x(3x-2)(4x+1)$

Answer:

$3x(3x - 2)(4x + 1)$