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Which term can be added to the list so that the greatest common factor of the three terms is [tex]\(12h^3\)[/tex]?

[tex]\(36h^3, 12h^6\)[/tex]

A. [tex]\(6h^3\)[/tex]

B. [tex]\(12h^2\)[/tex]

C. [tex]\(30h^4\)[/tex]

D. [tex]\(48h^5\)[/tex]


Sagot :

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]