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What is the G.C.F. (greatest common factor) of the terms of the expression [tex]$24 h^4 - 42 h^3 + 54 h^2$[/tex]?

A. [tex]$2 h$[/tex]
B. [tex]$4 h^2$[/tex]
C. [tex]$6 h^2$[/tex]
D. [tex]$10 h^3$[/tex]

Answer :

We need to find the greatest common factor (G.C.F.) of the expression

[tex]$$24h^4 - 42h^3 + 54h^2.$$[/tex]

Step 1: Find the G.C.F. of the numerical coefficients.

Consider the coefficients: 24, 42, and 54.

- Factorize each coefficient:
- [tex]\(24 = 2^3 \times 3\)[/tex],
- [tex]\(42 = 2 \times 3 \times 7\)[/tex],
- [tex]\(54 = 2 \times 3^3\)[/tex].

The common factors among these numbers are [tex]\(2\)[/tex] and [tex]\(3\)[/tex]. Multiplying these common factors, we have

[tex]$$2 \times 3 = 6.$$[/tex]

So, the G.C.F. of the coefficients is 6.

Step 2: Find the G.C.F. of the variable parts.

The variable parts in the terms are [tex]\(h^4\)[/tex], [tex]\(h^3\)[/tex], and [tex]\(h^2\)[/tex]. The smallest power of [tex]\(h\)[/tex] present in all the terms is [tex]\(h^2\)[/tex].

Step 3: Combine the results.

The overall G.C.F. is the product of the G.C.F. of the coefficients and the G.C.F. of the variable parts:

[tex]$$\text{G.C.F.} = 6h^2.$$[/tex]

Thus, the greatest common factor of the expression is

[tex]$$\boxed{6h^2}.$$[/tex]

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