Question

Use the Rational Zero Theorem to list all possible rational zeros for the given function.

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

Answer 1

The possible rational zeros are fractions of the form p/q where p is a factor of the last term 3 and q is a factor of the first coefficient 1 (of x^5)


So the possible rational zeros are: 3/1, -3/1, 1/1, -1/1


which reduce to: 3, -3, 1, -1

Answer 2

that's a neat theorem to learn. does this mean, since there are only 4 rational roots and we have a 5th degree polynomial, that the last root is imaginary/irrational?

Answer 3

no, imaginary roots come in pairs

so if the last root is imaginary, then you'll only have 3 rational roots (since you'll really have 2 imaginary roots)

Answer 4

and those are the POSSIBLE rational roots

Answer 5

D'oh! of course, so the last root must be irrational?
ah, I see from your last reply that the last root could also be rational

Answer 6

Ie, if there is a rational root, then it is one of those four values

Answer 7

it's perfectly possible to have 5 rational roots (you can easily have a double/triple root)

Answer 8

nowhere does it say we have to use each root once, those are just the only candidates we can use

Answer 9

Ok - I think I understand the principal now. thanks for the insight!

Answer 10

np

Answer 11

its a necessary condition , if f(x) = 0 where x= p/q , then it must be of the form above

Answer 12

then p divides the last term, and q divides the first term

Answer 13

but theres no guarantee there is any p/q such that f(p/q) = 0 . But if it exists , then it must be of the form above