A rational game 2
A polynomial with integer coefficients P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 0 , with a n and a 0 being coprime positive integers , has one of the roots 3 2 . Find the second smallest possible value of a 0 + a n .
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The answer is 7.
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1 solution
Can u explain rational roots theorem?
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It says that if q p is a root of polynomial, then p is factor of constant and q is factor of leading coefficient.
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Thank u very much. This set is useful
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@Shree Ganesh – Thanks for your comment that you found my set useful!
@Prince Loomba; If we take the constant term as -2 and the coefficient of leading term to be -3, then shouldn't the answer occur to be -5?
The fact that 'ao' and 'an' were coprime had a lot of meaning too as it eliminated the possibility that any one of them can be zero. Note that this was possible as every integer divides zero so rational root theorem would not be violated.
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Rightly said, but doesn't the question mention that a 0 and a n are positive integers?
For the sake of completeness, you should show that there does exist a polynomial P ( x ) with a n = 3 and a 0 = 4 such that P ( x ) has 3 2 as one of its roots. Finding such a polynomial is not hard ( 3 x 2 − 8 x + 4 is one such example). But not showing it leaves the solution slightly incomplete.
Rational root can be utilised here
Relevant wiki: Rational Root Theorem - Problem Solving
a 0 = 2 x and a n = 3 y because according to rational roots theorem, 2 is factor of a 0 and 3 is factor of a n . Thus we need second smallest value of 2 x + 3 y , which is obtained when x and y are 2 and 1 respectively. Hence answer is 4 + 3 = 7