Question

According to the rational root theorem what are all the potential rational roots of f(x)=5x^3-7x+11​

Answer 1

This is the only real root because roots are x-intercepts and there's only one x-intercept. Sometimes there would be three roots for a third degree polynomial but in this case it doesn't. The other roots are imaginary or complex numbers. So the only rational root is -1.653.

Answer 2

`±(1)/(1) , ±(1)/(11) , ±(5)/(1) , ±(5)/(11)`

Explanation:

If P(x) is a polynomial and we have to find all the potential rational roots of P(x) , we take all the possible ratios of the factors of "leading coefficient and the "constant term".

If
`P(x)=a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+a_(n-3)x^(n-3)......... a_o`

Possible Rational Roots

=±factors of
`a_n`/factors of
`a_o`

Here in our polynomial

`a_n= 5`

factors of 5 = 1 , 5

`a_o=11`

factors of 11 = 1,11

Hence possible rational roots are

±factors of 5 / factors of 11

±
`(1)/(1)` , ±
`(1)/(11)` , ±
`(5)/(1)` , ±
`(5)/(11)`