2010年10月17日 星期日

Polynomial roots

An elementary problem seen from other site.

Let be a polynomial with odd integral coefficients,

show that it cannot have rational root.

5 則留言:

tobywhcheng 提到...

related to Newton polytope?

Pop 提到...

You may use similar method of proving root 2 is not a rational number to solve this problem.

harryyau 提到...

the aim is to prove the sq root of delta is irrational ?
actually i can not solve this problem by the method contradiction...

Pop 提到...

Suppose there is a rational root x=p/q, where p, q are relatively prime integer.

put x=p/q into the polynomial.

Note: a, b and c are odd integral coeff.

finally check the parity.

MA 提到...

Suppose LHS can be factorized into (Ax+B)(Cx+D), where A, B, C & D are integers.
Then since a & c are odd, all A, B, C & D must be odd.
Expand to get the coeff. of x = AD+BC =b
b is odd but AD+BC is even.
Therefore, contradiction.