Question

How do you use the rational root theorem to find the roots of `f(x)=x^5-3x^2-4`?

Answer 1

(as far as I can tell) `f(x)=x^5-3x^2-4` does not have any rational roots (and therefore the Rational Root Theorem is of no use in determining the roots)

Explanation:

According to the Rational Root Theorem:
any rational roots of `color(red)(1)x^5-3x^2-color(blue)(4)`
must be of the form:
`color(white)("XXXX")("integer factor of "color(blue)(4))/("integer factor of "color(red)(1))`

The only possible rational roots are therefore:
`color(white)("XXXX") +-1, +-2, +-4`

`f(+1) = -6`
`f(-1) = -8`
`f(+2)=16`
`f(-2) =-48`
`f(+4)=972`
`f(-4)=-1076`
Since none of these give a value for `f(x)` which is equal to `0`
all possible rational roots provided by the Rational Root Theorem are extraneous.