This problem concerns polynomials, divisibility, and a similar equivalence relat

Question

This problem concerns polynomials, divisibility, and a similar equivalence relation to that above. A polynomial of degree n in a variable x wit?h real coefficients is the set of expressions of the form
f(x)=(j=0)^n ajx^j =a0+a1x+…+anx^n
Where n{0,1,2,…}=N, ajR for j{0,1,2,…,n} and an is not equal to 0}. The set of all polynomials is the set R[x]={(j=0)^n ajx^j =a0+a1x+…+anx^n :nN, ajR for j{0,1,2,…,n} and an is not equal to 0}
Show that degree defines a partial order on the set of polynomials. That is say that f(x)< g(x) if and only if deg(f(x))=deg(g(x)). Now show that the relation < is a reflexive, symmetric, transitive relation on R[x]. Now what is the set of polynomials of degree 0?

Explanation / Answer

A partial order is a binary relation "=" over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that: a = a (reflexivity); if a = b and b = a then a = b (antisymmetry); if a = b and b = c then a = c (transitivity). f(x)

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