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'll start by analyzing the structure and coefficients of [tex]\( g(x) \)[/tex].
Step 1: Analyze the given function [tex]\( g(x) \)[/tex].
- The degree is 5 (highest power of [tex]\( x \)[/tex]).
- The leading coefficient is 3.
- The constant term is 12.
The Rational Root Theorem states that any rational root of a polynomial is of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (12) and [tex]\( q \)[/tex] is a factor of the leading coefficient (3).
Factors of 12 are: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
Factors of 3 are: [tex]\( \pm 1, \pm 3 \)[/tex].
So the potential rational roots are:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm 3, \pm \frac{4}{3}, \pm \frac{6}{3}, \pm 12. \][/tex]
Step 2: Compare the given functions to find one with the same set of potential rational roots.
Function 1: [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- The degree is 5.
- The leading coefficient is 3.
- The constant term is -12.
Here, the constant term is -12 (not the same as [tex]\( g(x) \)[/tex]), meaning the set of potential rational roots may differ if applied strictly.
Function 2: [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- The degree is 6 (not the same as [tex]\( g(x) \)[/tex]).
- The leading coefficient is 3.
- The constant term is 0.
Since the degrees are different, the functions can’t match in terms of the set of rational roots.
Function 3: [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- The degree is 5.
- The leading coefficient is 12 (not the same as [tex]\( g(x) \)[/tex]).
- The constant term is 3.
The leading coefficient and constant term differ from [tex]\( g(x) \)[/tex]; hence, the set of rational roots will be different.
Function 4: [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- The degree is 5.
- The leading coefficient is 12 (not the same as [tex]\( g(x) \)[/tex]).
- The constant term is 48.
Again, since both the leading coefficient and constant term differ, the sets of rational roots will be different.
Conclusion:
Among all the provided functions, none have precisely matching degree, leading coefficients, and constant terms comparable to [tex]\( g(x) \)[/tex]. Therefore, based on these criteria, there is no function among the given choices that satisfies having the same set of potential rational roots as [tex]\( g(x) \)[/tex].
Thus, the correct result based on this comparison is:
[tex]\[ 0 \][/tex]
This means none of the given functions have the rational root pattern as the function [tex]\( g(x) \)[/tex].
Step 1: Analyze the given function [tex]\( g(x) \)[/tex].
- The degree is 5 (highest power of [tex]\( x \)[/tex]).
- The leading coefficient is 3.
- The constant term is 12.
The Rational Root Theorem states that any rational root of a polynomial is of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (12) and [tex]\( q \)[/tex] is a factor of the leading coefficient (3).
Factors of 12 are: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
Factors of 3 are: [tex]\( \pm 1, \pm 3 \)[/tex].
So the potential rational roots are:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm 3, \pm \frac{4}{3}, \pm \frac{6}{3}, \pm 12. \][/tex]
Step 2: Compare the given functions to find one with the same set of potential rational roots.
Function 1: [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- The degree is 5.
- The leading coefficient is 3.
- The constant term is -12.
Here, the constant term is -12 (not the same as [tex]\( g(x) \)[/tex]), meaning the set of potential rational roots may differ if applied strictly.
Function 2: [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- The degree is 6 (not the same as [tex]\( g(x) \)[/tex]).
- The leading coefficient is 3.
- The constant term is 0.
Since the degrees are different, the functions can’t match in terms of the set of rational roots.
Function 3: [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- The degree is 5.
- The leading coefficient is 12 (not the same as [tex]\( g(x) \)[/tex]).
- The constant term is 3.
The leading coefficient and constant term differ from [tex]\( g(x) \)[/tex]; hence, the set of rational roots will be different.
Function 4: [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- The degree is 5.
- The leading coefficient is 12 (not the same as [tex]\( g(x) \)[/tex]).
- The constant term is 48.
Again, since both the leading coefficient and constant term differ, the sets of rational roots will be different.
Conclusion:
Among all the provided functions, none have precisely matching degree, leading coefficients, and constant terms comparable to [tex]\( g(x) \)[/tex]. Therefore, based on these criteria, there is no function among the given choices that satisfies having the same set of potential rational roots as [tex]\( g(x) \)[/tex].
Thus, the correct result based on this comparison is:
[tex]\[ 0 \][/tex]
This means none of the given functions have the rational root pattern as the function [tex]\( g(x) \)[/tex].
