Suppose a, b are two irrational numbers such that ab is rational and a+b is rational. Then a, b are the solution to a quadratic polynomial with integer coeffecients.

clealtAfforcewug

clealtAfforcewug

Answered question

2022-11-05

Suppose a , b are two irrational numbers such that ab is rational and a + b is rational. Then a , b are the solution to a quadratic polynomial with integer coeffecients.

Answer & Explanation

andytronicoh4t

andytronicoh4t

Beginner2022-11-06Added 18 answers

Hint:
( x a ) ( x b ) = x 2 ( a + b ) x + a b .
trumansoftjf0

trumansoftjf0

Beginner2022-11-07Added 5 answers

Suppose a are two irrational numbers such that a b is rational and a + b is rational. Then a are the solution to a quadratic polynomial with integer coefficients.
Proof:
Because a + b and a b are rational, from the definition of a rational number we can write
a + b = m n , and a b = p q ,
where m, n, p and q are integers. Now we can write
( x a ) ( x b ) = x 2 ( a + b ) x + a b = x 2 m n x + p q = 1 n q ( n q x 2 m q x + n p ) .
So we have a quadratic polynomial with integer coefficients ( n q, m q and n p), and obviously the roots of this polynomial are a and b. Therefore we have the proof.

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