When the algebraic expression 12y^3 + 12y^2 - 45y is factored completely, which of the fol… — Armed Services Vocational Aptitude Battery (ASVAB)

When the algebraic expression $12y^3 + 12y^2 - 45y$ is factored completely, which of the following expressions is produced?

The completely factored form is

3y(2y - 3)(2y + 5)

The correct answer accurately extracts the full GCF of

3y

and correctly factors the remaining trinomial into

(2y - 3)(2y + 5)

. Expanding this result confirms it equals

12y^3 + 12y^2 - 45y

Identify and extract the Greatest Common Factor (GCF) from all terms in

12y^3 + 12y^2 - 45y

The GCF is

3y

. Factoring it out leaves

3y(4y^2 + 4y - 15)

Extracting the GCF first simplifies the polynomial and is a required step for complete factorization.

Factor the remaining non-monic trinomial,

4y^2 + 4y - 15

, by finding two numbers that multiply to

a \times c

and add to

b

We need numbers that multiply to

(4)(-15) = -60

and add to

4

. The numbers are

10

and

-6

This prepares the trinomial for factoring by grouping.

Rewrite the middle term (

4y

) using these two numbers and factor by grouping.

4y^2 + 10y - 6y - 15 = 2y(2y + 5) - 3(2y + 5) = (2y - 3)(2y + 5)

Grouping extracts common binomial factors, breaking down the quadratic into its simplest linear binomials.

Combine the original GCF with the newly factored binomials.

3y(2y - 3)(2y + 5)

To represent the full and complete factorization of the original polynomial.

Factoring polynomials by first extracting the GCF and subsequently using the ac-method (or grouping) to factor a non-monic quadratic trinomial.

Question