Recall that Poly(R) denotes the ring of polynomial function f: RR from the reals to the...

 

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Recall that Poly(R) denotes the ring of polynomial function f: RR from the reals to the reals. (a) Let c ER be a fixed real number and define the function: formula Poly(R) → R by the (f(x)) = f(c). Verify that is a surjective ring homomorphism. (b) The kernel of turns out to be a principal ideal of Poly(R). Find a polynomial function g(r) that generates this ideal. (Hint: g(x) has degree 1. Use The Factor Theorem, a theorem that you saw in High School.) Recall that Poly(R) denotes the ring of polynomial function f: RR from the reals to the reals. (a) Let c ER be a fixed real number and define the function: formula Poly(R) → R by the (f(x)) = f(c). Verify that is a surjective ring homomorphism. (b) The kernel of turns out to be a principal ideal of Poly(R). Find a polynomial function g(r) that generates this ideal. (Hint: g(x) has degree 1. Use The Factor Theorem, a theorem that you saw in High School.)