2010 STEP 3 Q3iv

I assumed there exists positive integers such that

$C_q (x) \equiv C_r (x) C_s(x)$

However the roots are primitive hence roots can only appear on both sides of equation if q=r or q=s

if q=r it means

$C_s (x) \equiv 1$

Since the above statement implies equivalent to 1 for all x, can't I just find say that a primitive root of the LHS will cause the LHS to equal 0 hence the equivalent to statement does not hold - contradiction

In the mark scheme, it just says 'it is not possible for positive s' - what does this mean?

Yes

They're saying that "$C_s(X)$ is not identically 1 for any positive integer s", for the reason that you've given.

Spot on!

The thing with marking STEP questions is there are many ways of doing the same question, making producing mark schemes a very difficult thing to do. I don't find the way that mark scheme is phrased here is particularly clear or well justified. The explanation you've given is both equivalent to what they've said but also much better because you've given a reason why it's true.

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP