f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]....

f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]. show that f is

1) a homomorphism

2) Ker(f)=(x-2)R[x]

3) prove that R[x]/Ker(f) is an isomorphism with R.

(R in this case is the Reals so R[x]=a0+a1x+a1x^2...anx^n)

Expert Solution

While showing homomorphism you can consider polynomial p(x) with less degree than another polynomial q(x) or may be both have same degree.

For showing 3rd question's proof I have used first isomorphism theorem which is very popular in abstract algebra.

Colby Messinger answered 2 years ago

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