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3.1. Irreducible Polynomials 83 Let p(t) be a manic polynomial over Z. Show that, if r is a rational root of p(t), then r must ...
84 3. Factors and Zeros Show that f(t) is irreducible. (Eugen Netto, Mufhematische Annalen 48 (1897)). Let r be a positive inte ...
3.2. Strategies for Factoring Polynomials over Z 85 6t + u and vt - 20 are both integer multiples of the same linear polynomial. ...
86 3. Factors and Zeros Because of the symmetry of the coefficients, one might try to find two symmetrically related factors, ea ...
3.2. Strategies for Factoring Polynomials over Z 87 (0) 15 - 3t4 - t2 - 4t + 14 (p) t” - 98t4 + 1. Let f be a nonzero homogeneo ...
88 3. Factors and Zeros Show that each of the following polynomials is irreducible over Z, for some m. Can you deduce from this ...
3.2. Strategies for Factoring Polynomials over Z 89 E.30. A sequence of polynomials tin(t) is defined by the recurrence: ulJ(t) ...
90 3. Factors and Zeros (15) Show that UT + up + *.. + 26; _ (?/*+I+ 1+ y-(*+‘))(y” + 1+ y-“) - 3(t + 1) u1+ ‘u2 + *. * + U” - ( ...
3.3. Finding Integer and Rational Roots 91 (a) Show that the rational number a/b is a zero of q(t) if and only if cnan + c”-la ...
92 3. Factors and Zeros ~“-1 = br”-2 - urn-l c,,-2 = br,-3 - ar,,-2 ... c2 = bq - ar2 cl = bra - arl co = -are. (c) In (b), supp ...
3.3. Finding Integer and Rational Roots 93 Divide cl + bso by a. If the result is not an integer, a/b is not a zero. Otherwise, ...
94 3. Factors and Zeros We can check whether a is a zero of a polynomial q(t) over Z by evaluating q(a) by Horner’s Method. In ...
3.4. Locating Integer Roots: Modular Arithmetic 95 take an odd value. Similarly, if t is given an odd value, then the value of t ...
(^96) 3. Factors and Zeros is also a solution of the congruence q(t) q 0 (modp’). Find all the positive integers less than 100 ...
3.4. Locating Integer Roots: Modular Arithmetic 97 (c) Let c = clal + c2a2 + c3a3 + .-. + c,a,. For which positive values of th ...
98 3. Factors and Zeros (f) A second solution of the congruence t2 + 2 G 0 (mod 3) is t z 2 (mod 3). Find a solution of t2 + 2 - ...
3.4. Locating Integer Roots: Modular Arithmetic 99 and (‘i’)+(L:)=( kT.1) fork=1,3,5 ,..., p-2. The induction step follows easil ...
100 3. Factors and Zeros Verify that the unique solution of this is v G 4 (mod 9). For the next step, let h = 2 and u = 4, so th ...
3.5. Roots of Unity 101 3.5 Roots of Unity The factorization of t” - 1 was studied by C.F. Gauss (1777-1855)) who wanted to find ...
102 3. Factors and Zeros By a suitable pairing of linear factors over C of the polynomial P,(t), show that we can obtain a fact ...
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