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Answer :
To solve this problem, we need to use the Rational Root Theorem. This theorem helps us find possible rational roots of a polynomial by considering the ratios of the factors of its constant term and leading coefficient.
The polynomial we're given is [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12 \)[/tex]. For this polynomial:
- The leading coefficient is 3.
- The constant term is 12.
According to the Rational Root Theorem, the potential rational roots are the possible ratios between the factors of the constant term and the leading coefficient.
Let's check each function to see if they could have the same set of potential rational roots as [tex]\( g(x) \)[/tex].
1. [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- Leading coefficient is 3.
- Constant term is -12.
The constant terms are different (12 vs. -12), so [tex]\( f(x) \)[/tex] does not have the same set of potential rational roots.
2. [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- The degree is different (6, not 5), so it can't match [tex]\( g(x) \)[/tex].
The degrees are different, which means the structure isn't the same, so this doesn't work.
3. [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- Leading coefficient is 12.
- Constant term is 3.
The leading coefficients are different (3 vs. 12), so [tex]\( f(x) \)[/tex] does not have the same set of potential rational roots.
4. [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- Leading coefficient is 12.
- Constant term is 48.
Neither the leading coefficient nor the constant term matches [tex]\( g(x) \)[/tex], so this doesn't work either.
After checking all the options, none of the functions listed have the exact same leading coefficient, constant term, and degree as [tex]\( g(x) \)[/tex]. Therefore, none of the given functions have the same set of potential rational roots as the original polynomial [tex]\( g(x) \)[/tex].
The polynomial we're given is [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12 \)[/tex]. For this polynomial:
- The leading coefficient is 3.
- The constant term is 12.
According to the Rational Root Theorem, the potential rational roots are the possible ratios between the factors of the constant term and the leading coefficient.
Let's check each function to see if they could have the same set of potential rational roots as [tex]\( g(x) \)[/tex].
1. [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- Leading coefficient is 3.
- Constant term is -12.
The constant terms are different (12 vs. -12), so [tex]\( f(x) \)[/tex] does not have the same set of potential rational roots.
2. [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- The degree is different (6, not 5), so it can't match [tex]\( g(x) \)[/tex].
The degrees are different, which means the structure isn't the same, so this doesn't work.
3. [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- Leading coefficient is 12.
- Constant term is 3.
The leading coefficients are different (3 vs. 12), so [tex]\( f(x) \)[/tex] does not have the same set of potential rational roots.
4. [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- Leading coefficient is 12.
- Constant term is 48.
Neither the leading coefficient nor the constant term matches [tex]\( g(x) \)[/tex], so this doesn't work either.
After checking all the options, none of the functions listed have the exact same leading coefficient, constant term, and degree as [tex]\( g(x) \)[/tex]. Therefore, none of the given functions have the same set of potential rational roots as the original polynomial [tex]\( g(x) \)[/tex].
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