To determine which term can be added to maintain the greatest common factor (GCF) of [tex]$12h^3$[/tex] for the list, we need to consider the GCF of the first two given terms and compare it to the other candidates.
### Given Terms:
1. [tex]\( 36h^3 \)[/tex]
2. [tex]\( 12h^6 \)[/tex]
First, let's recall that the GCF of two constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is the largest integer that divides both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] without a remainder. The variable part [tex]\(h^k\)[/tex] will be the lowest power of [tex]\(h\)[/tex] that appears in both terms.
1. Finding the constant GCF of 36 and 12:
[tex]\[
\text{GCF}(36, 12) = 12
\][/tex]
2. Finding the variable part:
[tex]\[
\text{The variable part } h^{\min(3, 6)} = h^3
\][/tex]
So, the combined GCF of [tex]\(36h^3\)[/tex] and [tex]\(12h^6\)[/tex] is:
[tex]\[
12h^3
\][/tex]
We need to determine which of the given candidate terms maintains the GCF of [tex]\(12h^3\)[/tex].
### Candidates:
1. [tex]\( 6h^3 \)[/tex]
2. [tex]\( 12h^2 \)[/tex]
3. [tex]\( 30h^4 \)[/tex]
4. [tex]\( 48h^5 \)[/tex]
### Checking the Candidates:
1. Candidate: [tex]\(6h^3\)[/tex]
Constant GCF: GCF(12, 6) = 6 (not 12)
Variable part: [tex]\(h^3\)[/tex]
Combined GCF: [tex]\(6h^3\)[/tex]
2. Candidate: [tex]\(12h^2\)[/tex]
Constant GCF: GCF(12, 12) = 12
Variable part: [tex]\(h^{\min(3, 2)} = h^2\)[/tex]
Combined GCF: [tex]\(12h^2\)[/tex] (not [tex]\(12h^3\)[/tex])
3. Candidate: [tex]\(30h^4\)[/tex]
Constant GCF: GCF(12, 30) = 6 (not 12)
Variable part: [tex]\(h^3\)[/tex]
Combined GCF: [tex]\(6h^3\)[/tex]
4. Candidate: [tex]\(48h^5\)[/tex]
Constant GCF: GCF(12, 48) = 12
Variable part: [tex]\(h^{\min(3, 5)} = h^3\)[/tex]
Combined GCF: [tex]\(12h^3\)[/tex]
So, adding [tex]\(48h^5\)[/tex] maintains the GCF of the three terms as [tex]\(12h^3\)[/tex].
Therefore, the term that can be added to the list so that the greatest common factor of the three terms is [tex]\(12h^3\)[/tex] is:
[tex]\[
\boxed{48h^5}
\][/tex]
