Question

Problem 7. Assume that a subset S of polynomials with real coefficients has a property:
If polynomials a(x), b(x) are from S and n(x), m(x) are any two polynomials with real coefficients, then polynomial a(x)n(x) + m(x)n(x) is again in S. Prove that there is a polynomial d(x) from S, such that any other polynomial from S is a multiple of d(x).

Answer 1

7. Given be subset of polynomials with real coefficients and has a property ,

If and are any two polynomial with real coefficient then .

Define degree of that is M is the collection of degree's of polynimials of .

By well ordering principal every subset of has a least element .

has a least element , say and be the polynomial in of degree .

we will prove that every polynomial of is multiple of  .

Let .

If is not a multiple of then by division algorithm there exist a polynomial with such that ,

Now , so by the property of ,

  

a contradiction to is the least degree polynomial in as and .

So our assumption   is not a multiple of is wrong .

Hence every elements of is multiple of .

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If you have any doubt or need more clarification at any step please comment .