To find the greatest common factor (GCF) of the numerator of the expression [tex]\(\frac{2x^3 - 3x^2 + 12x}{3x}\)[/tex], we need to focus on the polynomial in the numerator, which is [tex]\(2x^3 - 3x^2 + 12x\)[/tex].
Here are the steps involved:
1. Identify the Terms in the Numerator:
The terms in the numerator are:
[tex]\[
2x^3, \quad -3x^2, \quad 12x
\][/tex]
2. Factor Each Term:
- The term [tex]\(2x^3\)[/tex] can be factored as [tex]\(2 \cdot x \cdot x \cdot x\)[/tex].
- The term [tex]\(-3x^2\)[/tex] can be factored as [tex]\(-3 \cdot x \cdot x\)[/tex].
- The term [tex]\(12x\)[/tex] can be factored as [tex]\(12 \cdot x\)[/tex].
3. Identify the Common Factors:
The factors for each term are:
[tex]\[
2x^3: \quad 2, x, x, x
\][/tex]
[tex]\[
-3x^2: \quad -3, x, x
\][/tex]
[tex]\[
12x: \quad 12, x
\][/tex]
The only factor that is common to all terms is [tex]\(x\)[/tex].
4. Greatest Common Factor for the Coefficients:
Let's consider the numerical coefficients: 2, -3, and 12. The greatest common divisor (GCD) of these coefficients is 1 because they do not share any common factors other than 1.
5. Combine the Common Factors:
The GCF is the highest degree of [tex]\(x\)[/tex] that divides all terms. As seen, the common [tex]\(x\)[/tex] term in all three expressions is [tex]\(x\)[/tex].
Therefore, the GCF of the numerator [tex]\(2x^3 - 3x^2 + 12x\)[/tex] is [tex]\(x\)[/tex].
So the correct answer is:
[tex]\[
C. \, 3 x
\][/tex]
But considering only the greatest common factor of the terms excluding 3 from the factors in entire.
The correct answer should be:
[tex]\[
\text{None from the above answers if considered 3 as common factor in entire however if without 3 will be correct as X}\][/tex]
