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Answer :
We begin by considering the Rational Root Theorem. For the polynomial
[tex]$$
g(x)=3x^5-2x^4+9x^3-x^2+12,
$$[/tex]
the theorem tells us that any potential rational root can be written as
[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.
Here, the constant term is [tex]$12$[/tex], and the leading coefficient is [tex]$3$[/tex]. Therefore, the potential rational roots are given by
[tex]$$
\pm \frac{p}{q} \quad \text{with} \quad p \text{ dividing } 12 \quad \text{and} \quad q \text{ dividing } 3.
$$[/tex]
This means the set of potential rational roots for [tex]$g(x)$[/tex] is determined by the factors of [tex]$12$[/tex] (namely, [tex]$1, 2, 3, 4, 6, 12$[/tex]) and the factors of [tex]$3$[/tex] (namely, [tex]$1, 3$[/tex]).
Now, we need to identify which of the following functions has the same set of potential rational roots as [tex]$g(x)$[/tex].
1) [tex]$$f(x)=3x^5-2x^4-9x^3+x^2-12$$[/tex]
2) [tex]$$f(x)=3x^6-2x^5+9x^4-x^3+12x$$[/tex]
3) [tex]$$f(x)=12x^5-2x^4+9x^3-x^2+3$$[/tex]
4) [tex]$$f(x)=12x^5-8x^4+36x^3-4x^2+48$$[/tex]
Let’s analyze each one:
1. For [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] (its absolute value is [tex]$12$[/tex]).
Since the factors of the leading coefficient and the constant term are the same as those for [tex]$g(x)$[/tex], the potential rational roots for this function are exactly the same as those of [tex]$g(x)$[/tex].
2. For [tex]$$f(x)=3x^6-2x^5+9x^4-x^3+12x,$$[/tex]
- Notice the constant term is missing (effectively, it is [tex]$0$[/tex]), which changes the set of potential rational roots. This does not match [tex]$g(x)$[/tex].
3. For [tex]$$f(x)=12x^5-2x^4+9x^3-x^2+3,$$[/tex]
- The leading coefficient here is [tex]$12$[/tex].
- The constant term is [tex]$3$[/tex] (its absolute value is [tex]$3$[/tex]).
The factors of the leading coefficient and the constant term are different from those of [tex]$g(x)$[/tex], so the set of potential rational roots will differ.
4. For [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] (its absolute value is [tex]$48$[/tex]).
Again, the factors are different from [tex]$3$[/tex] and [tex]$12$[/tex], resulting in a different set of potential rational roots.
Since only option 1 has a leading coefficient of [tex]$3$[/tex] and a constant term (in absolute value) of [tex]$12$[/tex], it follows that the set of potential rational roots for option 1 is the same as that for [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]
The answer is: Option 1.
[tex]$$
g(x)=3x^5-2x^4+9x^3-x^2+12,
$$[/tex]
the theorem tells us that any potential rational root can be written as
[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.
Here, the constant term is [tex]$12$[/tex], and the leading coefficient is [tex]$3$[/tex]. Therefore, the potential rational roots are given by
[tex]$$
\pm \frac{p}{q} \quad \text{with} \quad p \text{ dividing } 12 \quad \text{and} \quad q \text{ dividing } 3.
$$[/tex]
This means the set of potential rational roots for [tex]$g(x)$[/tex] is determined by the factors of [tex]$12$[/tex] (namely, [tex]$1, 2, 3, 4, 6, 12$[/tex]) and the factors of [tex]$3$[/tex] (namely, [tex]$1, 3$[/tex]).
Now, we need to identify which of the following functions has the same set of potential rational roots as [tex]$g(x)$[/tex].
1) [tex]$$f(x)=3x^5-2x^4-9x^3+x^2-12$$[/tex]
2) [tex]$$f(x)=3x^6-2x^5+9x^4-x^3+12x$$[/tex]
3) [tex]$$f(x)=12x^5-2x^4+9x^3-x^2+3$$[/tex]
4) [tex]$$f(x)=12x^5-8x^4+36x^3-4x^2+48$$[/tex]
Let’s analyze each one:
1. For [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] (its absolute value is [tex]$12$[/tex]).
Since the factors of the leading coefficient and the constant term are the same as those for [tex]$g(x)$[/tex], the potential rational roots for this function are exactly the same as those of [tex]$g(x)$[/tex].
2. For [tex]$$f(x)=3x^6-2x^5+9x^4-x^3+12x,$$[/tex]
- Notice the constant term is missing (effectively, it is [tex]$0$[/tex]), which changes the set of potential rational roots. This does not match [tex]$g(x)$[/tex].
3. For [tex]$$f(x)=12x^5-2x^4+9x^3-x^2+3,$$[/tex]
- The leading coefficient here is [tex]$12$[/tex].
- The constant term is [tex]$3$[/tex] (its absolute value is [tex]$3$[/tex]).
The factors of the leading coefficient and the constant term are different from those of [tex]$g(x)$[/tex], so the set of potential rational roots will differ.
4. For [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] (its absolute value is [tex]$48$[/tex]).
Again, the factors are different from [tex]$3$[/tex] and [tex]$12$[/tex], resulting in a different set of potential rational roots.
Since only option 1 has a leading coefficient of [tex]$3$[/tex] and a constant term (in absolute value) of [tex]$12$[/tex], it follows that the set of potential rational roots for option 1 is the same as that for [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]
The answer is: Option 1.
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