An engineer needs to simplify a calculation involving the volume of a component, modeled b… — Armed Services Vocational Aptitude Battery (ASVAB)

An engineer needs to simplify a calculation involving the volume of a component, modeled by the polynomial $4y^3 + 12y^2 - 72y$ . What is the fully factored equivalent of this polynomial?

$4y(y + 6)(y - 3)$ Answer
B
$4y(y - 6)(y + 3)$
C
$4y(y^2 + 3y - 18)$
D
$4y(y + 6)(y + 3)$

Answer

4y(y + 6)(y - 3)

To fully factor the given expression, first extract the greatest common factor (GCF). All terms share a common factor of

4y

, yielding

4y(y^2 + 3y - 18)

. Next, factor the resulting quadratic trinomial

y^2 + 3y - 18

. The two numbers that multiply to

-18

and sum to

3

are

6

and

-3

. This gives the binomial factors

(y + 6)

and

(y - 3)

. Combining these with the GCF results in the completely factored form

4y(y + 6)(y - 3)

Step-by-Step Solution

Identify and extract the greatest common factor (GCF) from all terms in the polynomial

4y^3 + 12y^2 - 72y

4y(y^2 + 3y - 18)

Factoring out the GCF simplifies the remaining polynomial, making it easier to factor completely.

Factor the remaining quadratic trinomial

y^2 + 3y - 18

by finding two integers that multiply to

-18

and add to

3

(y + 6)(y - 3)

The integers

6

and

-3

satisfy both conditions:

6 \times (-3) = -18

and

6 + (-3) = 3

Combine the GCF with the factored binomials to state the final expression.

4y(y + 6)(y - 3)

This represents the original polynomial decomposed entirely into its lowest degree factors.

Key Concept

Factoring Polynomials with a GCF

Estimated Time:1m 0s

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