Question

need answer as soon as possible. thanks

Consider the ring Rix) of polynomials with real coefficients, with operations polynomial addition and polynomial multiplication (you don't have to prove this is a ring). For example, for the polynomials f(x)=1+2x+3x2 and g(x)=3-5x, we have f(x)+g(x)= (1+2x+3x2)+(3-5x)-4-3x+3x2 and f(x)g(x)(1+2x+3x2)(3-5x)=3+X-X2-15x). Show that the function h: RIX-R given by h(f(x)=f(0) is a ring homomorphism. Then describe the kernel ker(h).

Answer 1

h: R[X] given by b. (76)) = f(0) to show: _b is a ring homomorphism. Let Fon of gou are in R[a] consider h (fa) +9 ()) = n(f+9(N) = f+9.00) = FO) +960) = hefc) thgow) h[f@2E9%)] = b[fa] thiago) Also, h[fa.g(x)] = b[196] = $9.00) = f(o)-9602 _ = hful h@c_022 i bisa ring hemomczehisme Nent to find kernel of h kech : {FC) e R[a] Lhrfa)] = 0 F _= potart--tanxmelcall 90=0 " fol= 0 = Cha=0 → Iresh - f Q, at a222 4 - fanno l aier =<ng . Kerh =<a7.