We present three theorems involving rational numbers.
th Power Theorem. Let be a positive integer. If is a positive integer such that for some rational number , then must be an integer.
Proof: We apply the integer root theorem to the polynomial . Since this polynomial has a rational root , this root must be an integer.
Theorem 2. If the sum and product of two rational numbers are both integers, then the two rational numbers must be integers.
Proof: We have and , hence are rational roots of the polynomial . By the integer root theorem, and are both integers.
Theorem 3. A quadratic polynomial has integer coefficients. Then both of the roots are integers if and only if:
1. is a perfect square,
2. divides both and .
Proof: By the Remainder Factor Theorem, if the roots are integers and , then . By comparing terms, we obtain and . Hence, (2) is satisfied. Furthermore, so (1) is also satisfied.
Conversely, by the quadratic formula, the roots of are . By condition (1), it follows that the roots are rational. By condition (2) and Vieta's formula, both the sum and product of the roots are integers. Hence by Theorem 2, the roots are integers.