Here\'s the problem statement: f(x) is a quadratic polynomial with integer coeff
ID: 2962624 • Letter: H
Question
Here's the problem statement:
f(x) is a quadratic polynomial with integer coefficients and a nonzero constant term. Using Lemma 1, prove that if f(x) has an irrational root, theb bot roots of f(x) are irrational. You may utilize familiar facts about quadratic polynomials from algebra as needed.
So I know I have to use lemma 1. The lemma is: "The sum of a rational number and an irrational number is irrational".
Here's the proof of lemma 1:
We will proceed by contradiction. Let r be a rational number, and i be an irrational. Now, suppose their sum was rational, say s. Since r is rational there are two integers p and q such that r = p/q. Now if the the sum of r and i were rational, then there would be two integers t and u such that s = t/u. That means that
r + i = s
p/q + i = t/u
i = t/u - p/q
i = tq/qu - pu/qu
i = (tq -pu)/qu
But (tq -pu)/qu is rational by definition, which means i would be rational, but i is irrational, hence the contradiction. So, the sum of a rational and an irrational is always irrational.
Explanation / Answer
GIVEN