Answer :
To determine which function has the same set of potential rational roots as the function [tex]\(g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12\)[/tex], we can use the Rational Root Theorem. This theorem tells us that any rational root, expressed as a fraction [tex]\(\frac{p}{q}\)[/tex], must have [tex]\(p\)[/tex] as a factor of the constant term and [tex]\(q\)[/tex] as a factor of the leading coefficient.
For [tex]\(g(x)\)[/tex]:
- The constant term is [tex]\(12\)[/tex].
- The leading coefficient is [tex]\(3\)[/tex].
We need to compare the constant terms and leading coefficients of the other functions to find which one has the same potential rational roots.
Let's analyze the options:
1. [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex]
- Leading coefficient: [tex]\(3\)[/tex]
- Constant term: [tex]\(-12\)[/tex]
2. [tex]\(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)[/tex]
- Leading coefficient: [tex]\(3\)[/tex]
- Constant term: N/A (the polynomial does not have a constant term)
3. [tex]\(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)[/tex]
- Leading coefficient: [tex]\(12\)[/tex]
- Constant term: [tex]\(3\)[/tex]
4. [tex]\(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)[/tex]
- Leading coefficient: [tex]\(12\)[/tex]
- Constant term: [tex]\(48\)[/tex]
To have the same set of potential rational roots as [tex]\(g(x)\)[/tex], a function must match both the leading coefficient and the constant term. None of the listed options exactly matches both the leading coefficient ([tex]\(3\)[/tex] for [tex]\(g(x)\)[/tex]) and the constant term ([tex]\(12\)[/tex] for [tex]\(g(x)\)[/tex]).
Thus, none of the given functions have the same set of potential rational roots as [tex]\(g(x)\)[/tex]. The result concludes that the function with the same potential rational roots does not exist among the provided options.
For [tex]\(g(x)\)[/tex]:
- The constant term is [tex]\(12\)[/tex].
- The leading coefficient is [tex]\(3\)[/tex].
We need to compare the constant terms and leading coefficients of the other functions to find which one has the same potential rational roots.
Let's analyze the options:
1. [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex]
- Leading coefficient: [tex]\(3\)[/tex]
- Constant term: [tex]\(-12\)[/tex]
2. [tex]\(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)[/tex]
- Leading coefficient: [tex]\(3\)[/tex]
- Constant term: N/A (the polynomial does not have a constant term)
3. [tex]\(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)[/tex]
- Leading coefficient: [tex]\(12\)[/tex]
- Constant term: [tex]\(3\)[/tex]
4. [tex]\(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)[/tex]
- Leading coefficient: [tex]\(12\)[/tex]
- Constant term: [tex]\(48\)[/tex]
To have the same set of potential rational roots as [tex]\(g(x)\)[/tex], a function must match both the leading coefficient and the constant term. None of the listed options exactly matches both the leading coefficient ([tex]\(3\)[/tex] for [tex]\(g(x)\)[/tex]) and the constant term ([tex]\(12\)[/tex] for [tex]\(g(x)\)[/tex]).
Thus, none of the given functions have the same set of potential rational roots as [tex]\(g(x)\)[/tex]. The result concludes that the function with the same potential rational roots does not exist among the provided options.
