7. = = = Let fi(x) = Anxn + An-1.xn-1 + ... + a1x + do, f2(x) = bnxn + bn-1.xn-1 + ... + b1x + bo, anX + , g1(x) = CnX"

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7 Let Fi X Anxn An 1 Xn 1 A1x Do F2 X Bnxn Bn 1 Xn 1 B1x Bo Anx G1 X Cnx 1 (81.74 KiB) Viewed 21 times

7. = = = Let fi(x) = Anxn + An-1.xn-1 + ... + a1x + do, f2(x) = bnxn + bn-1.xn-1 + ... + b1x + bo, anX + , g1(x) = CnX" + Cn-1.xn-1 + ... + C1X + Co, g2(x) = dụx" + dn-1Xn-1 + ... + dıx + do , ) f(x) = knX" + kn-1.xn-1 + ... + kix + ko, g(x) = tnx" + tn-1Xn-1 + ... + fix + to. x- Two polynomials are said to be congruent as polynomials modulo m if for each power of x, the coefficients of that power in both polynomials are congruent modulo m. In symbols, suppose the two polynomials are f(x) and g(x), then f(x) =m g(x). Show that (a) 11x3 + x2 + 2 = x3 – 4x2 + 5x + 22. (b) if f(x) =m g(x), then f(a)=m g(a) for any integer a. (c) even if f(a)=m g(a) for any integer a, f(x) and g(x) may not be congruent as polynomials modulo m. [Hint: Think of m and p(x) such that p(x) = f(x) – g(x) and m1p(x).] (d) if fi(x)=m gi(x) and f2(x)=m g2(x), then (i) (f1 + f2)(x) =m (g1 + g2)(x) and (ii) (f1f2)(x) =m (8182)(x) (e) if h(x) = x(x + 1)(2x + 1), then + (i) h(a)=6 0 for all integers a, and (ii) verify the values of a such that (x – a) is NOT a factor of h(x), where -2 sas3. (f) if f(a) =m 0, then there is a polynomial q(x) with integral coefficients ri - al such that f(x) =m (x – a) q(x) [Hint: Simplify ..] X - a