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To determine which function has the same set of potential rational roots as the function [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex], we need to examine the leading coefficients and the constant terms of each polynomial.
The Rational Root Theorem states that any potential rational root of a polynomial [tex]\( a_n x^n + \cdots + a_1 x + a_0 \)[/tex] is of the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex] and [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].
For [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex]:
- The leading coefficient is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 12 \)[/tex].
Now, let's analyze each given function:
1. [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( -12 \)[/tex]
2. [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( 0 \)[/tex]
3. [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 3 \)[/tex]
4. [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 48 \)[/tex]
By examining the leading coefficients and the constant terms, we can see that the potential rational roots are determined by the factors of these terms.
Comparing with [tex]\( g(x) \)[/tex]:
- The polynomial [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex] has the same leading coefficient [tex]\( 3 \)[/tex] and its constant term [tex]\( -12 \)[/tex] shares the same set of factors as [tex]\( 12 \)[/tex].
- The polynomial [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex] also has the leading coefficient [tex]\( 3 \)[/tex] but has a constant term of [tex]\( 0 \)[/tex], which does not share the same factors as [tex]\( 12 \)[/tex].
- The polynomial [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex] has a different leading coefficient [tex]\( 12 \)[/tex] and a different constant term [tex]\( 3 \)[/tex].
- The polynomial [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex] also has a different leading coefficient [tex]\( 12 \)[/tex] and a different constant term [tex]\( 48 \)[/tex].
Thus, the polynomial with the same set of potential rational roots as [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex] is:
[tex]\[ f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \][/tex]
And also:
[tex]\[ f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \][/tex]
The Rational Root Theorem states that any potential rational root of a polynomial [tex]\( a_n x^n + \cdots + a_1 x + a_0 \)[/tex] is of the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex] and [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].
For [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex]:
- The leading coefficient is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 12 \)[/tex].
Now, let's analyze each given function:
1. [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( -12 \)[/tex]
2. [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]
- Leading coefficient: [tex]\( 3 \)[/tex]
- Constant term: [tex]\( 0 \)[/tex]
3. [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 3 \)[/tex]
4. [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]
- Leading coefficient: [tex]\( 12 \)[/tex]
- Constant term: [tex]\( 48 \)[/tex]
By examining the leading coefficients and the constant terms, we can see that the potential rational roots are determined by the factors of these terms.
Comparing with [tex]\( g(x) \)[/tex]:
- The polynomial [tex]\( f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex] has the same leading coefficient [tex]\( 3 \)[/tex] and its constant term [tex]\( -12 \)[/tex] shares the same set of factors as [tex]\( 12 \)[/tex].
- The polynomial [tex]\( f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex] also has the leading coefficient [tex]\( 3 \)[/tex] but has a constant term of [tex]\( 0 \)[/tex], which does not share the same factors as [tex]\( 12 \)[/tex].
- The polynomial [tex]\( f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex] has a different leading coefficient [tex]\( 12 \)[/tex] and a different constant term [tex]\( 3 \)[/tex].
- The polynomial [tex]\( f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex] also has a different leading coefficient [tex]\( 12 \)[/tex] and a different constant term [tex]\( 48 \)[/tex].
Thus, the polynomial with the same set of potential rational roots as [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex] is:
[tex]\[ f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \][/tex]
And also:
[tex]\[ f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \][/tex]
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