Irrational + Irrational = Rational?
Algebra
Level
2
True or false :
The sum of the roots of a polynomial of second degree with integer coefficients is always rational .
True
False
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1 solution
Adding the two roots given by the quadratic formula, we see that the term under the radical cancels out.
2 a − b + b 2 − 4 a c + 2 a − b − b 2 − 4 a c = 2 ∗ − b / 2 a = − b / a
Since b and a are integers, the sum of the roots can be represented as a fraction, and thus it is rational.
Bonus question : What about polynomials of 3rd degree?