Arithmetic of Polynomials. if k is a field and f,g,h are polynomials such that f=g+h, then a polynomial that divides two of the three will divide the third.

Evelyn Freeman

Evelyn Freeman

Answered question

2022-10-25

Arithmetic of Polynomials
if k is a field and f,g,h are polynomials such that f = g + h, then a polynomial that divides two of the three will divide the third.
I'm having trouble showing doing the problem as polynomials. I know how to do it is f,g,h are integers.

Answer & Explanation

faux0101d

faux0101d

Beginner2022-10-26Added 21 answers

Step 1
Do it the usual way (as for integers):
If p is a polynomial over k dividing both g,h this means that g p , h p are also polynomials over k.
Step 2
Thus:
f p = g + h p = g p + h p
is also a polynomial over k. This means that p also divides f.

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