Answer :
To determine which function has the same set of potential rational roots as [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12 \)[/tex], we'll use the Rational Root Theorem. This theorem states that any potential rational root of a polynomial function is a ratio [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.
For the polynomial [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex]:
1. Identify the constant term and leading coefficient:
- The constant term ([tex]\( c \)[/tex]) is [tex]\( 12 \)[/tex].
- The leading coefficient ([tex]\( a \)[/tex]) is [tex]\( 3 \)[/tex].
2. Find the factors of the constant term [tex]\( 12 \)[/tex]:
- Factors of [tex]\( 12 \)[/tex] are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Find the factors of the leading coefficient [tex]\( 3 \)[/tex]:
- Factors of [tex]\( 3 \)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
4. Form the set of potential rational roots:
- The potential rational roots are ratios of the factors of the constant term to the factors of the leading coefficient.
- These potential roots are: [tex]\( \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{3}{1}, \pm \frac{4}{1}, \pm \frac{6}{1}, \pm \frac{12}{1}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \)[/tex].
Now, we need to compare these potential rational roots with those for each of the given functions:
### For [tex]\( f_1(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( -12 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 3 \)[/tex].
2. Find the factors of the constant term [tex]\( -12 \)[/tex]:
- Factors of [tex]\( -12 \)[/tex] are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Factors of the leading coefficient [tex]\( 3 \)[/tex]:
- Factors of [tex]\( 3 \)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
4. Potential rational roots:
- Ratios of these factors are: [tex]\( \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{3}{1}, \pm \frac{4}{1}, \pm \frac{6}{1}, \pm \frac{12}{1}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \)[/tex].
These potential roots match those of the initial polynomial [tex]\( g(x) \)[/tex].
### For [tex]\( f_2(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 0 \)[/tex] (implies at least one rational root is [tex]\( 0 \)[/tex]).
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 3 \)[/tex].
However, given that the root [tex]\( 0 \)[/tex] is present, the potential rational roots set will include [tex]\( 0 \)[/tex], differing significantly from [tex]\( g(x) \)[/tex]'s set.
### For [tex]\( f_3(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 3 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 12 \)[/tex].
### For [tex]\( f_4(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 48 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 12 \)[/tex].
After evaluating the potential rational roots for all polynomial choices, we see that the polynomial [tex]\( f_1(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex] has the same set of potential rational roots as [tex]\( g(x) \)[/tex].
Thus, the function with the same set of potential rational roots as [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12 \)[/tex] is:
[tex]\[ f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \][/tex].
For the polynomial [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 0x + 12 \)[/tex]:
1. Identify the constant term and leading coefficient:
- The constant term ([tex]\( c \)[/tex]) is [tex]\( 12 \)[/tex].
- The leading coefficient ([tex]\( a \)[/tex]) is [tex]\( 3 \)[/tex].
2. Find the factors of the constant term [tex]\( 12 \)[/tex]:
- Factors of [tex]\( 12 \)[/tex] are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Find the factors of the leading coefficient [tex]\( 3 \)[/tex]:
- Factors of [tex]\( 3 \)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
4. Form the set of potential rational roots:
- The potential rational roots are ratios of the factors of the constant term to the factors of the leading coefficient.
- These potential roots are: [tex]\( \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{3}{1}, \pm \frac{4}{1}, \pm \frac{6}{1}, \pm \frac{12}{1}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \)[/tex].
Now, we need to compare these potential rational roots with those for each of the given functions:
### For [tex]\( f_1(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( -12 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 3 \)[/tex].
2. Find the factors of the constant term [tex]\( -12 \)[/tex]:
- Factors of [tex]\( -12 \)[/tex] are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Factors of the leading coefficient [tex]\( 3 \)[/tex]:
- Factors of [tex]\( 3 \)[/tex] are [tex]\( \pm 1, \pm 3 \)[/tex].
4. Potential rational roots:
- Ratios of these factors are: [tex]\( \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{3}{1}, \pm \frac{4}{1}, \pm \frac{6}{1}, \pm \frac{12}{1}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \)[/tex].
These potential roots match those of the initial polynomial [tex]\( g(x) \)[/tex].
### For [tex]\( f_2(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 0 \)[/tex] (implies at least one rational root is [tex]\( 0 \)[/tex]).
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 3 \)[/tex].
However, given that the root [tex]\( 0 \)[/tex] is present, the potential rational roots set will include [tex]\( 0 \)[/tex], differing significantly from [tex]\( g(x) \)[/tex]'s set.
### For [tex]\( f_3(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 3 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 12 \)[/tex].
### For [tex]\( f_4(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48 \)[/tex]:
1. Identify the constant term and leading coefficient:
- Constant term ([tex]\( c \)[/tex]): [tex]\( 48 \)[/tex].
- Leading coefficient ([tex]\( a \)[/tex]): [tex]\( 12 \)[/tex].
After evaluating the potential rational roots for all polynomial choices, we see that the polynomial [tex]\( f_1(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \)[/tex] has the same set of potential rational roots as [tex]\( g(x) \)[/tex].
Thus, the function with the same set of potential rational roots as [tex]\( g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12 \)[/tex] is:
[tex]\[ f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12 \][/tex].
