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Polynomial Sprint extra question -2

Prove that if the co-efficients of the quadratic equation \(ax^{2}+bx+c = 0\) are odd integers,the roots of the equations cannot be rational numbers.

Note by Eddie The Head
2 years, 6 months ago

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Let a = 2d + 1, b = 2e + 1, c = 2f + 1 where d,e,f are integers.

Using the quadratic formula we get x = (-b +- sqrt{b^2 - 4ac})/(2a). We will show that b^2 - 4ac cannot be a square number.

Substituting our above equations (a = 2d + 1, b = 2e + 1, c = 2f + 1) into b^2 - 4ac we get 4e^2 + 4e - 16df - 8d - 8f - 3. We can rewrite this as (8)(-2df - d - f) + (4)(e)(e + 1) - 3. Since one of (e) and (e+1) must be even, it follows that 4(e)(e+1) must be divisible by 8. Therefore b^2 - 4ac is congruent to 5 modulo 8. This means that b^2 - 4ac is not a square number, as square numbers must be congruent to 0, 1, or 4 mod 8. This shows that x is irrational. Colin Tang · 2 years, 6 months ago

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Suppose that p/q is a root of the quadratic equation and that p,q are coprime, then c is divisible by p and a is divisible by q. Hence both p and q are odd. Substituting p/q for x, we get: \(\frac{ap^2}{q^2} + \frac{p}{q} + c = 0 \). However, this is clearly impossible as both p, q are odd. Hence there are no rational roots for the equation. Jianzhi Wang · 2 years, 6 months ago

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@Jianzhi Wang There are 3 odd terms in the final equation. If all of them are positive, then we are done. If one is negative, then the LHS will still be an odd number (odd + odd - odd = odd != 0). Jianzhi Wang · 2 years, 6 months ago

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@Jianzhi Wang Why is it impossible? Eddie The Head · 2 years, 6 months ago

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@Jianzhi Wang Could you clarify please? Why is this "clearly impossible"? You can rewrite it as ((a/q)(p)(p) + p)/q = -c where a/q is an odd integer (a is divisible by q, a and q are both odd) and p and q and c are all odd integers--how do you know that the fraction isn't simplifiable to -c

Never mind, I think I've figured it out. Since a/q is odd and p is odd, (a/q)(p)(p) is odd and p is odd, which makes (a/q)(p)(p) + p even. You want (a/q)(p)(p) + p to equal -cq, which is a product of two odd integers and therefore odd. But you can't have an even number equal to an odd number. Colin Tang · 2 years, 6 months ago

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@Colin Tang He means that if we multiply by \(q^2\), LHS will be odd and RHS =0, clearly a contradiction. Bogdan Simeonov · 2 years, 6 months ago

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@Bogdan Simeonov If the coefficients are odd integers, they can be negative, right? We could somehow choose the coefficients so that LHS is zero. However, if we write the equation as \(cq^{2} = -(ap^{2} + bpq)\) and see different cases by taking \(p, q\) to be even or odd, we see that the equation may be satisfied only when both \(p, q\) are even. But that contradicts the condition that \(p, q\) are coprime.

This is what Colin says. Siladitya Basu · 2 years, 6 months ago

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@Bogdan Simeonov Ok that makes sense now Thanks Colin Tang · 2 years, 6 months ago

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