By

Professor

I am having lots of problems with section nine. Can you tell me what types of questions I should concentrate on?

For example, I don't even know where to start with question 9.20(below)

For polynomials f(x) and g(x) with real coefficients, let f(x)~g(x) mean that f'(x)=g'(x) (where the primes denote derivatives). Prove that ~ is an equivalence realtion, and give a complete set of equivalence class representatives. (A polynomial with real coefficients is a expression of the form a0 + a1x + ...+ anx(to the power n), where a0, a1,...an belong to R

I don't know where to start - can you give me a pointer on what I should do when it comes to proving congruences.

Thank you for your time

By

Start by showing this is an equivalence. Here is a bit of it:

1. (Reflexive) Is f~f for every polynomial f? Sure it is, since f'=f' for every polynomial f.

2. (Symmetric relation?) Suppose f~g. Then f'=g'. So g'=f'. So g~f.

3. (Transitivity) I am leaving this to you.

The equivalence classes: {f(x)+c; c in R}; that is, one equivalence class is the set consisting of a polynomial and all other polynomials that have graphs paralel to that polynomial.