Answered: Let f(x) = a,x" + a, p* + ...+ a, E…

Let f(x) = a,x" + a, p* + ...+ a, E Z[x] and a, + 0. Prove that if r and s are relatively prime integers and f(r/s) = 0, then r|a, and s | a,.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let
f(x) = a,x" + a, p*
+ ...+ a, E Z[x]
and a, + 0. Prove that if r and s are relatively prime integers and
f(r/s) = 0, then r|a, and s | a,.

Expert Solution

Step 1

It is given that, $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{0} \in ℤ [x]$ with $a_{n} \neq 0$ .

Also, it is given that $r$ and $s$ are relatively prime integers and $f (\frac{r}{s}) = 0$ .

Then, we have to prove that $r a_{0}$ and $s a_{n}$ .

Step by step

Solved in 2 steps

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