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Answer :
We start with the polynomial
[tex]$$
g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12.
$$[/tex]
The Rational Root Theorem tells us that any rational root of a polynomial with integer coefficients must be of the form
[tex]$$
\pm \frac{p}{q},
$$[/tex]
where [tex]$p$[/tex] is a factor of the constant term and [tex]$q$[/tex] is a factor of the leading coefficient.
Step 1. Identify Factors
1. The constant term in [tex]$g(x)$[/tex] is [tex]$12$[/tex]. Its factors are:
[tex]$$
\pm 1,\, \pm 2,\, \pm 3,\, \pm 4,\, \pm 6,\, \pm 12.
$$[/tex]
2. The leading coefficient is [tex]$3$[/tex]. Its factors are:
[tex]$$
\pm 1,\, \pm 3.
$$[/tex]
Step 2. List All Possible Rational Roots for [tex]$g(x)$[/tex]
Using the Rational Root Theorem, the possible rational roots are obtained by forming the fractions
[tex]$$
\pm \frac{p}{q} \quad \text{with} \quad p \text{ a factor of } 12 \text{ and } q \text{ a factor of } 3.
$$[/tex]
Thus, the potential roots are
[tex]$$
\pm 1,\, \pm 2,\, \pm 3,\, \pm 4,\, \pm 6,\, \pm 12,\, \pm \frac{1}{3},\, \pm \frac{2}{3},\, \pm \frac{4}{3}.
$$[/tex]
This collection of roots is the complete set predicted by the theorem for [tex]$g(x)$[/tex].
Step 3. Analyze the Answer Options
Let's check which function among the given options has the same set of potential rational roots by comparing the factors of the constant term and the leading coefficient.
1. Option 1:
[tex]$$
f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12.
$$[/tex]
- The leading coefficient is [tex]$3$[/tex].
- The constant term is [tex]$-12$[/tex] (the sign does not affect the factors considered).
The factors for the constant term are the same as for [tex]$g(x)$[/tex] ([tex]$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$[/tex]) and the factors for the leading coefficient remain [tex]$\pm 1, \pm 3$[/tex]. Hence, the possible rational roots for this function are identical to those of [tex]$g(x)$[/tex].
2. Option 2:
[tex]$$
f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x.
$$[/tex]
Here the constant term is [tex]$0$[/tex] (since the term free of [tex]$x$[/tex] is missing), which means the possible rational roots will include [tex]$0$[/tex] and have a different set of candidates.
3. Option 3:
[tex]$$
f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3.
$$[/tex]
- The leading coefficient is [tex]$12$[/tex].
- The constant term is [tex]$3$[/tex].
The factors here differ from those in [tex]$g(x)$[/tex] ([tex]$\pm 1, \pm 3$[/tex] for the constant and [tex]$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$[/tex] for the leading coefficient would be reversed), so the set of possible rational roots will be different.
4. Option 4:
[tex]$$
f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48.
$$[/tex]
- The leading coefficient is [tex]$12$[/tex].
- The constant term is [tex]$48$[/tex].
Again, since these factors are different, the set of potential rational roots will not match those for [tex]$g(x)$[/tex].
Conclusion
Only Option 1 maintains a leading coefficient of [tex]$3$[/tex] and a constant term of [tex]$-12$[/tex], which guarantees the same set of potential rational roots as
[tex]$$
\pm \frac{p}{q} \text{ with } p \in \{1,2,3,4,6,12\} \text{ and } q \in \{1,3\}.
$$[/tex]
Thus, the function that has the same set of potential rational roots as [tex]$g(x)$[/tex] is
[tex]$$
\boxed{f(x)=3x^5-2x^4-9x^3+x^2-12.}
$$[/tex]
[tex]$$
g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12.
$$[/tex]
The Rational Root Theorem tells us that any rational root of a polynomial with integer coefficients must be of the form
[tex]$$
\pm \frac{p}{q},
$$[/tex]
where [tex]$p$[/tex] is a factor of the constant term and [tex]$q$[/tex] is a factor of the leading coefficient.
Step 1. Identify Factors
1. The constant term in [tex]$g(x)$[/tex] is [tex]$12$[/tex]. Its factors are:
[tex]$$
\pm 1,\, \pm 2,\, \pm 3,\, \pm 4,\, \pm 6,\, \pm 12.
$$[/tex]
2. The leading coefficient is [tex]$3$[/tex]. Its factors are:
[tex]$$
\pm 1,\, \pm 3.
$$[/tex]
Step 2. List All Possible Rational Roots for [tex]$g(x)$[/tex]
Using the Rational Root Theorem, the possible rational roots are obtained by forming the fractions
[tex]$$
\pm \frac{p}{q} \quad \text{with} \quad p \text{ a factor of } 12 \text{ and } q \text{ a factor of } 3.
$$[/tex]
Thus, the potential roots are
[tex]$$
\pm 1,\, \pm 2,\, \pm 3,\, \pm 4,\, \pm 6,\, \pm 12,\, \pm \frac{1}{3},\, \pm \frac{2}{3},\, \pm \frac{4}{3}.
$$[/tex]
This collection of roots is the complete set predicted by the theorem for [tex]$g(x)$[/tex].
Step 3. Analyze the Answer Options
Let's check which function among the given options has the same set of potential rational roots by comparing the factors of the constant term and the leading coefficient.
1. Option 1:
[tex]$$
f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12.
$$[/tex]
- The leading coefficient is [tex]$3$[/tex].
- The constant term is [tex]$-12$[/tex] (the sign does not affect the factors considered).
The factors for the constant term are the same as for [tex]$g(x)$[/tex] ([tex]$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$[/tex]) and the factors for the leading coefficient remain [tex]$\pm 1, \pm 3$[/tex]. Hence, the possible rational roots for this function are identical to those of [tex]$g(x)$[/tex].
2. Option 2:
[tex]$$
f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x.
$$[/tex]
Here the constant term is [tex]$0$[/tex] (since the term free of [tex]$x$[/tex] is missing), which means the possible rational roots will include [tex]$0$[/tex] and have a different set of candidates.
3. Option 3:
[tex]$$
f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3.
$$[/tex]
- The leading coefficient is [tex]$12$[/tex].
- The constant term is [tex]$3$[/tex].
The factors here differ from those in [tex]$g(x)$[/tex] ([tex]$\pm 1, \pm 3$[/tex] for the constant and [tex]$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$[/tex] for the leading coefficient would be reversed), so the set of possible rational roots will be different.
4. Option 4:
[tex]$$
f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48.
$$[/tex]
- The leading coefficient is [tex]$12$[/tex].
- The constant term is [tex]$48$[/tex].
Again, since these factors are different, the set of potential rational roots will not match those for [tex]$g(x)$[/tex].
Conclusion
Only Option 1 maintains a leading coefficient of [tex]$3$[/tex] and a constant term of [tex]$-12$[/tex], which guarantees the same set of potential rational roots as
[tex]$$
\pm \frac{p}{q} \text{ with } p \in \{1,2,3,4,6,12\} \text{ and } q \in \{1,3\}.
$$[/tex]
Thus, the function that has the same set of potential rational roots as [tex]$g(x)$[/tex] is
[tex]$$
\boxed{f(x)=3x^5-2x^4-9x^3+x^2-12.}
$$[/tex]
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