How to find all of the possible rational roots of the polynomial function?

Question

How to find all of the possible rational roots of the polynomial function?

f(x) = 6x^4 -4x^3 -3x + 2

How would I do this problem?

Answer 1

In this case, you can notice that your polynomial can be factored as follows:

 

6x⁴-4x³-3x+2=(6x⁴-3x)-(4x³-2)=3x(2x³-1)-2(2x³-1)=(3x-2)(2x³-1)

 

The roots of your polynomial are either the roots of the first term, or the second. The root of the first term is 2/3. The second term boils down to:

 

2x³=1 or x=(1/2)^1/3

 

Obviously, cube root of 1/2 is irrational (two other roots are complex), thus the only rational root in this case is x=2/3

Answer 2

The Rational Root/Zero Theorem states that IF there are rational roots/zeros of the form p/q, with no common factors, then p will be a factor of the constant term and q will be a factor of the leading coefficient.

 

f(x) = 6x^4 -4x^3 -3x + 2

 

p would be a factor of 2: ±1, ±2

q would be a factor of 6: ±1, ±2, ±3, ±6

 

p/q would be one of these: ± 1/6, ± 1/3, ± 1/2, ± 2/3, ± 1, ± 2

 

That's the list of possible rational roots.