Prove that 2x^4 + x^2 - x + 1 has no rational zeros.

Question

I understand that the above polynomial has no rational roots, but I'm too sure on how to prove this. Originally I just tested P(2), P(1) and P(1/2), and none of them gave the answer of zero - thus these are not roots of the polynomial. Is this all I need to do or am I required to provide further evidence and if so, how should I answer the question?

Answer 1

Use the Rational Root Theorem.  

 

If p/q is a rational root of the given polynomial, then p is a divisor of the constant term, 1, and q is a divisor of the leading coefficient, 2.

 

So, p = 1 or -1 and q = 1, -1, 2, or -2

 

Therefore, the only possibilities that need to be checked are 

1, -1, 1/2, and -1/2.

 

Check each of the four possibilities (by plugging in or by Synthetic Division) to show that none of the possibilities is a root.