I'm having trouble with this specific problem at the moment. The theorem states that if n/m is a rational root of a polynomial with integer coefficients, the leading coefficient is divisible by m and the free coefficient is divisible by n.
Using this theorem, I'm supposed to prove that √1+3√2 is irrational.
I don't have any idea where to start on this one.
Any help or hints are appreciated.
Answer
You want to use the rational root theorem.
Hint: Let x=√1+3√2, then, x2=1+3√2, so (x2−1)=3√2. Hence, (x2−1)3=2.
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