Question

What are the possible rational roots of `6x^4+35x^3-x^2-7x-1=0` and then determine the rational roots?

Answer 1

The possible rational roots are `+-1;+-1/2;+-1/3;+-1/6`;
The rational roots are `1/2;-1/3`

Explanation:

The possible rational roots are obtained by dividing each of the factors of the known term (-1) by each of the factors of the maximum degree term's coefficient (6).

Then they are:

`+-1;+-1/2;+-1/3;+-1/6`

To determine the rational roots, let's use the remainder rule:

`color(red)(x=1)-> 6+35-1-7-1=32!=0`
`color(red)(x=-1)-> 6-35-1+7-1=-24!=0`
`color(red)(x=1/2): 6(1/2)^4+35(1/2)^3-(1/2)^2-7(1/2)-1=0`

Then `x=1/2` is a rational root.

You also can use a spreadsheet to find zeros faster.

Then you would find only another rational root `x=-1/3`