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

Mathematics June 10, 2019 Sabyasachi Sub ring Criterion (necessary and sufficient conditions for sub ring): Let S be a non empty subset of a ring R. Then S is a sub ring of R if and only if 1. a, bE S a - bE S 2. a,bE S abE S Solution (a). R is the set of all polynomials p(x) of R of R[a Let a,bE R. Then a,bE Rr also. such that deg(p(r)) = 0. Clearly R is a non empty subset R[ and Ris a ring, so a -bE R. Therefore, a - bE R 1. Since a, b 2. Since a, b R[x and R is a ring, so ab E Rx. Therefore, ab E R. Therefore, R is a sub ring of R[x]. (b). No, R is not a sub ring of Ra]. Let p(x), (x) R{x]k such that deg(p(x)) = k and deg(g(x)) = k. Then deg((p)(x))= kk =2k. So, (pq)(x) R (Although, (p+q)(x) € R(z]k. ) Therefore, Ra& is not a sub ring of Ra (c). Note that, 2x 64 = 1225 = in Z11. -2x4 +624 +3 -5 -824 -8443 +3 4a +3a 5 43 +222 6x4 4a322- 2 2x1 -2x2 +3r -5 -22 -ar 4ac -5 4a +2 7 Note that, -7 = -7 11 = 4 in Z11 64-43 2r - 2 Therefore the quotient is and the remainder is 4 (d). Z2 {0,1. A general 3-degree polynomial in Z2 is: ax3ba2+cx +d, where a,b, c, d E Z2 and a O. a has only one choice 1 and b, e, d has two choices 0 and 1. So there are 1 x 2 x 2 x 2 = 8 elements of degree 3 in Z2. They are, (1) 3 (2) x3 2 (3) x3 +x (4) x3 + x2 + & (5) x3 +1 (6) a2 1 (7) x3 + a2 + 1 (8) x* + x + 1