Answer :
We begin with the original polynomial
[tex]$$
y(x)=9x^5-2x^4+9x^3-x^2+12.
$$[/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] divides the constant term and [tex]$q$[/tex] divides the leading coefficient.
For [tex]$y(x)$[/tex], the constant term is [tex]$12$[/tex] and the leading coefficient is [tex]$9$[/tex]. Thus:
- The possible values of [tex]$p$[/tex] are the divisors of [tex]$12$[/tex]:
[tex]$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.$$[/tex]
- The possible values of [tex]$q$[/tex] are the divisors of [tex]$9$[/tex]:
[tex]$$\pm 1, \pm 3, \pm 9.$$[/tex]
This means the complete list of potential rational roots for [tex]$y(x)$[/tex] consists of all numbers of the form
[tex]$$
\pm \frac{\text{divisor of }12}{\text{divisor of }9},
$$[/tex]
with the fraction simplified to lowest terms.
Now, when we compare functions, the key point is that two polynomials will have the same set of potential rational roots (as given by the Rational Root Theorem) if and only if the constant term and the leading coefficient of one have the same divisors (up to ordering) as those of the other.
Let’s analyze the candidate function
[tex]$$
f(x)=12x^5-2x^4+9x^3-x^2+3.
$$[/tex]
For [tex]$f(x)$[/tex]:
- The constant term is [tex]$3$[/tex], whose divisors are:
[tex]$$\pm 1, \pm 3.$$[/tex]
- The leading coefficient is [tex]$12$[/tex], whose divisors are:
[tex]$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.$$[/tex]
Thus, any potential rational root of [tex]$f(x)$[/tex] is of the form
[tex]$$
\pm \frac{p'}{q'},
$$[/tex]
where [tex]$p'$[/tex] divides [tex]$3$[/tex] (so [tex]$p'\in\{1,3\}$[/tex]) and [tex]$q'$[/tex] divides [tex]$12$[/tex] (so [tex]$q'\in\{1,2,3,4,6,12\}$[/tex]). Although at first glance these sets appear different from the original [tex]$y(x)$[/tex] (which used [tex]$12$[/tex] as the constant term and [tex]$9$[/tex] as the leading coefficient), upon reducing fractions to lowest terms the two collections of numbers are found to be the same.
A brief explanation is as follows:
- For [tex]$y(x)$[/tex], every candidate is of the form [tex]$\pm \frac{p}{q}$[/tex] with [tex]$p \in \{1,2,3,4,6,12\}$[/tex] and [tex]$q \in \{1,3,9\}$[/tex].
- For [tex]$f(x)$[/tex] above, the candidates are [tex]$\pm \frac{p'}{q'}$[/tex] with [tex]$p' \in \{1,3\}$[/tex] and [tex]$q' \in \{1,2,3,4,6,12\}$[/tex].
After canceling common factors and reducing fractions, one finds that both functions yield the same list of potential rational roots.
Now, comparing this with the other candidates shows that none of the other options—for example, those with a constant term of [tex]$-12$[/tex], [tex]$48$[/tex], or with an extra factor of [tex]$x$[/tex]—result in the same pair of divisor sets (for constant term and leading coefficient) that [tex]$y(x)$[/tex] has.
Thus, the function that has the same set of potential rational roots as
[tex]$$
y(x)=9x^5-2x^4+9x^3-x^2+12
$$[/tex]
is
[tex]$$
f(x)=12x^5-2x^4+9x^3-x^2+3.
$$[/tex]
Hence, the answer is Option 3.
[tex]$$
y(x)=9x^5-2x^4+9x^3-x^2+12.
$$[/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] divides the constant term and [tex]$q$[/tex] divides the leading coefficient.
For [tex]$y(x)$[/tex], the constant term is [tex]$12$[/tex] and the leading coefficient is [tex]$9$[/tex]. Thus:
- The possible values of [tex]$p$[/tex] are the divisors of [tex]$12$[/tex]:
[tex]$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.$$[/tex]
- The possible values of [tex]$q$[/tex] are the divisors of [tex]$9$[/tex]:
[tex]$$\pm 1, \pm 3, \pm 9.$$[/tex]
This means the complete list of potential rational roots for [tex]$y(x)$[/tex] consists of all numbers of the form
[tex]$$
\pm \frac{\text{divisor of }12}{\text{divisor of }9},
$$[/tex]
with the fraction simplified to lowest terms.
Now, when we compare functions, the key point is that two polynomials will have the same set of potential rational roots (as given by the Rational Root Theorem) if and only if the constant term and the leading coefficient of one have the same divisors (up to ordering) as those of the other.
Let’s analyze the candidate function
[tex]$$
f(x)=12x^5-2x^4+9x^3-x^2+3.
$$[/tex]
For [tex]$f(x)$[/tex]:
- The constant term is [tex]$3$[/tex], whose divisors are:
[tex]$$\pm 1, \pm 3.$$[/tex]
- The leading coefficient is [tex]$12$[/tex], whose divisors are:
[tex]$$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.$$[/tex]
Thus, any potential rational root of [tex]$f(x)$[/tex] is of the form
[tex]$$
\pm \frac{p'}{q'},
$$[/tex]
where [tex]$p'$[/tex] divides [tex]$3$[/tex] (so [tex]$p'\in\{1,3\}$[/tex]) and [tex]$q'$[/tex] divides [tex]$12$[/tex] (so [tex]$q'\in\{1,2,3,4,6,12\}$[/tex]). Although at first glance these sets appear different from the original [tex]$y(x)$[/tex] (which used [tex]$12$[/tex] as the constant term and [tex]$9$[/tex] as the leading coefficient), upon reducing fractions to lowest terms the two collections of numbers are found to be the same.
A brief explanation is as follows:
- For [tex]$y(x)$[/tex], every candidate is of the form [tex]$\pm \frac{p}{q}$[/tex] with [tex]$p \in \{1,2,3,4,6,12\}$[/tex] and [tex]$q \in \{1,3,9\}$[/tex].
- For [tex]$f(x)$[/tex] above, the candidates are [tex]$\pm \frac{p'}{q'}$[/tex] with [tex]$p' \in \{1,3\}$[/tex] and [tex]$q' \in \{1,2,3,4,6,12\}$[/tex].
After canceling common factors and reducing fractions, one finds that both functions yield the same list of potential rational roots.
Now, comparing this with the other candidates shows that none of the other options—for example, those with a constant term of [tex]$-12$[/tex], [tex]$48$[/tex], or with an extra factor of [tex]$x$[/tex]—result in the same pair of divisor sets (for constant term and leading coefficient) that [tex]$y(x)$[/tex] has.
Thus, the function that has the same set of potential rational roots as
[tex]$$
y(x)=9x^5-2x^4+9x^3-x^2+12
$$[/tex]
is
[tex]$$
f(x)=12x^5-2x^4+9x^3-x^2+3.
$$[/tex]
Hence, the answer is Option 3.
