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 states that any rational root of a polynomial, expressed in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers, must be such that [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
### Step-by-Step Solution
1. Identify the leading coefficient and constant term of [tex]\( g(x) \)[/tex]:
- The leading coefficient (coefficient of [tex]\( x^5 \)[/tex]) is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 12 \)[/tex].
2. List the factors of the leading coefficient ([tex]\( 3 \)[/tex]) and the constant term ([tex]\( 12 \)[/tex]):
- Factors of [tex]\( 3 \)[/tex]: [tex]\( \pm 1, \pm 3 \)[/tex]
- Factors of [tex]\( 12 \)[/tex]: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
3. Form all possible rational roots [tex]\( \frac{p}{q} \)[/tex]:
- Possible rational roots are [tex]\( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 4, \pm \frac{4}{3}, \pm 6, \pm \frac{6}{3}, \pm 12, \pm \frac{12}{3} \)[/tex]
Simplifying those:
- [tex]\( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12 \)[/tex]
4. Apply the Rational Root Theorem to each candidate polynomial:
Let's check which polynomial among the given options has the same set of potential rational roots.
- For [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]:
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( -12 \)[/tex]
- The potential rational roots will be the same [tex]\( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12 \)[/tex]
- For [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]:
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( 0 \)[/tex]
- Potential rational roots need to include [tex]\( 0 \)[/tex] and others will be filtered by different configuration.
- For [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]:
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 3 \)[/tex]
- Factors of [tex]\( 12 \)[/tex] and [tex]\( 3 \)[/tex] don't match [tex]\( 3 \)[/tex] and [tex]\( 12 \)[/tex] of original function [tex]\( \pm 1, \pm \frac{1}{12}, \pm other \)[/tex].
- For [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]:
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 48 \)[/tex]
- Different combinations needing verification.
After considering correctly:
To verify completely.
Therefore, the function
[tex]\[ \boxed{3x^5 - 2x^4 - 9x^3 + x^2 - 12} \][/tex]
has the same set of potential rational roots as [tex]\[g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12.\][/tex]
Key Function that matches all is:
[tex]\(\boxed{1}\)[/tex]
