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Answer :
Sure! Let's go through how to determine which function has the same set of potential rational roots as the given polynomial [tex]\(3x^5 - 2x^4 + 9x^3 - x^2 + 12\)[/tex], using the Rational Root Theorem.
### Rational Root Theorem
The Rational Root Theorem states that any potential 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, and [tex]\(q\)[/tex] is a factor of the leading coefficient.
### Given Polynomial
The polynomial provided is:
[tex]\[3x^5 - 2x^4 + 9x^3 - x^2 + 12\][/tex]
- Constant Term: [tex]\(12\)[/tex]
- Leading Coefficient: [tex]\(3\)[/tex]
#### Factors:
- Factors of the constant term [tex]\(12\)[/tex] are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\)[/tex].
- Factors of the leading coefficient [tex]\(3\)[/tex] are [tex]\(\pm 1, \pm 3\)[/tex].
### Potential Rational Roots
By using these factors, the potential rational roots for the polynomial are all possible fractions formed by the factors of the constant term over the factors of the leading coefficient:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 1, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 2, \pm 12, \pm 4
\][/tex]
### Options Given
Let's determine the potential rational roots for each given polynomial and compare them.
#### Comparing Options:
We'll summarize the calculation results for each option:
1. Option 1: [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex] has the same leading coefficient and constant term.
2. Option 2: [tex]\(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)[/tex] has the same leading coefficient and no constant term, so roots differ.
3. Option 3: [tex]\(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)[/tex]
- Constant Term: 3
- Leading Coefficient: 12
- Potential rational roots are different.
4. Option 4: [tex]\(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)[/tex]
- Constant Term: 48
- Leading Coefficient: 12
- Potential rational roots are different.
### Conclusion
After evaluating the potential rational roots for each option, the polynomial [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex] has the same set of potential rational roots as the given polynomial [tex]\(3x^5 - 2x^4 + 9x^3 - x^2 + 12\)[/tex] because it shares the same constant term and leading coefficient. Therefore, the correct option is:
[tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex].
### Rational Root Theorem
The Rational Root Theorem states that any potential 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, and [tex]\(q\)[/tex] is a factor of the leading coefficient.
### Given Polynomial
The polynomial provided is:
[tex]\[3x^5 - 2x^4 + 9x^3 - x^2 + 12\][/tex]
- Constant Term: [tex]\(12\)[/tex]
- Leading Coefficient: [tex]\(3\)[/tex]
#### Factors:
- Factors of the constant term [tex]\(12\)[/tex] are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\)[/tex].
- Factors of the leading coefficient [tex]\(3\)[/tex] are [tex]\(\pm 1, \pm 3\)[/tex].
### Potential Rational Roots
By using these factors, the potential rational roots for the polynomial are all possible fractions formed by the factors of the constant term over the factors of the leading coefficient:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 1, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 2, \pm 12, \pm 4
\][/tex]
### Options Given
Let's determine the potential rational roots for each given polynomial and compare them.
#### Comparing Options:
We'll summarize the calculation results for each option:
1. Option 1: [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex] has the same leading coefficient and constant term.
2. Option 2: [tex]\(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)[/tex] has the same leading coefficient and no constant term, so roots differ.
3. Option 3: [tex]\(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)[/tex]
- Constant Term: 3
- Leading Coefficient: 12
- Potential rational roots are different.
4. Option 4: [tex]\(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)[/tex]
- Constant Term: 48
- Leading Coefficient: 12
- Potential rational roots are different.
### Conclusion
After evaluating the potential rational roots for each option, the polynomial [tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex] has the same set of potential rational roots as the given polynomial [tex]\(3x^5 - 2x^4 + 9x^3 - x^2 + 12\)[/tex] because it shares the same constant term and leading coefficient. Therefore, the correct option is:
[tex]\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)[/tex].
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