Answer Happy

Activity 12.6. Up to this point, we have only used long division for polynomials over the real numbers R. However, we can also use the same process for dividing polynomials over any field F. In this case, all computations with the coefficients must be done in the field F. For example, in Z3 [x], we could ask the question, "Does g(x) = [2]x² +x+ [1] divide ƒ(x) = x¹ + x³ + [2]x²+x+ [2]?” We can start the long division process as follows: [2]x²+x+ [1]) and [2].x² x¹ + x³+[2]x²+x+[2] x¹+[2]x³+[2]x² 2x³ +x+[2] Remember that all of the above calculations are being performed in Z3 [x], and so [2]x² ([2]x² + x + [1]) = [4]x:ª + [2]2:³ + [2]x² = x² + [2]x³ + [2].2², x² + x³ + [2]x² + x + [2] − (x² + [2]x³ + [2]x²) = ([1] − [1]) x² + ([1] − [2]) x³ + ([2] − [2]) x² + x + [2] = [2]x³ + x + [2]. (a) Complete this long division process to find polynomials q(r) and r(x) in Z3[x] such that f(x) = g(x)q(x) +r(x) and 0 ≤ deg(r(x)) < 2. Does g(x) divide f(x) in Z3 [x]? (b) In Z5[r], let g(x) = [2]x² + x + [1] and f(x) = x² + x³ + [2]x² + x + [2]. Use long division to find polynomials q(r) and r(r) in Z5 [x] such that f(x) = g(x)q(x) +r(x) and 0 ≤ deg(r(x)) < 2. Does g(x) divide f(x) in Z5[x]?