Question

(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z x].

Answer 1

R is the set of all polynomials p(x) of R [x] such that deg (p(x)) = 0. Clearly r is a non empty subset of R[x]. Let a, b elementof R. Then a, b elementof R [x] also. Since a, b elementof R[x] and R[x] is a ring, so a - b elementof R[x]. Therefore, a - b elementof R. Since a, b elementof R[x] and R[x] is a ring, so ab elementof R[x]. Therefore, ab elementof R. Therefore, R is a sub ring R[x]. No, R[x]_k is not a sub ring of R[x]. Let p(x), q(x) elementof R[x]_k such that deg (p(x)) = k and deg (q(x)) = k. Then deg ((pq)(x)) = k + k = 2k. So, (pq) (x) NotElementof R[x]_k. (Although, (p + q)(x) elementof R[x]_k.) Therefore, R[x]_k is not a sub ring of R[x]. Note that, 2x middot 6x^4 = 12x^5 = x^5 in Z_11. Note that, -7