Question

According to the Rational Root Theorem, which function has the same set of potential rational roots as the function g(x) = 3x5 – 2x4 + 9x3 – x2 + 12?

A).f(x) = 3x5 – 2x4 – 9x3 + x2 – 12
B).f(x) = 3x6 – 2x5 + 9x4 – x3 + 12x
C).f(x) = 12x5 – 2x4 + 9x3 – x2 + 3
D).f(x) = 12x5 – 8x4 + 36x3 – 4x2 + 48

Answer 1

According to the Rational Root Theorem, f(x) = 3x^5 – 2x^4 – 9x^3 + x^2 – 12 has the same set of potential rational roots as the function g(x) = 3x^5 – 2x^4 + 9x^3 – x^2 + 12

Answer 2

We have to identify the function which has the same set of potential rational roots as the function
`g(x)= 3x^5-2x^4+9x^3-x^2+12`.

Firstly, we will find the rational roots of the given function.

Let 'p' be the factors of 12

So, p=
`\pm 1, \pm 2, \pm 3, \pm 4, \pm 6`

Let 'q' be the factors of 3

So, q=
`\pm 1, \pm 3`

So, the rational roots are given by
`(p)/(q)` which are as:

`\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm (1)/(3), \pm (2)/(3), \pm (4)/(3)`.

Consider the first function given in part A.

f(x) =
`3x^5-2x^4-9x^3+x^2-12`

Here also, Let 'p' be the factors of 12

So, p=
`\pm 1, \pm 2, \pm 3, \pm 4, \pm 6`

Let 'q' be the factors of 3

So, q=
`\pm 1, \pm 3`

So, the rational roots are given by
`(p)/(q)` which are as:

`\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm (1)/(3), \pm (2)/(3), \pm (4)/(3)`.

Therefore, this equation has same rational roots of the given function.

Option A is the correct answer.