Math CentralQuandaries & Queries


Question from Nazrul, a teacher:

If (x+ a) be the H,C.F. of x^2+px+q and x^2+mx+n, prove that (p-m)a=q-n

I proved the above problem as follows:

Let f(x)= x^2+px+q and g(x)= x^2+mx+n
Therefore, (x+ a) be the H,C.F. of f(x) and g(x).
Therefore, (x+ a) be the common factor of f(x) and g(x).
Therefore, f(-a)=0 and g(-a)=0.
Therefore, f(-a)= g(-a)
or, (-a)^2+p(-a)+q =(-a)^2+m(-a)+n
or, a^2-pa+q =a^2-ma+n
or, q-n= a^2-ma-a^2+pa
or, q-n= pa-ma
or, q-n= (p-m)a
Therefore, (p-m)a= q-n

Is the above proscess correct? A teacher told that the proscess is wrong. If wrong please explain why the proscess is wrong and how I am to prve it.

Nazrul, I see nothing wrong at all in your proof. It appears to me to be complete for the set of complex numbers.

There are a number of ways of proving this and your method is a good one. The teacher who said your "process is wrong" must want you to use a different (particular) method, but the method you have used is also valid.

You will have to ask the teacher which process he or she wants you to use instead.

Stephen La Rocque

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