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Chapter 8 Solutions
Elements Of Modern Algebra
- 1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. andarrow_forwardFor any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π Φd (x), where the product runs overall positive divisors d of n greater than 1.arrow_forwardFind a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic polynomial g(x) of least degree with real coefficients that has the given numbers as zeros. 3 - i, i, and 2arrow_forward
- In the field GF(pn), show that for every positive divisor d of n,xpn- x has an irreducible factor over GF(p) of degree d.arrow_forwardFind the wronskian of f1=x4, f2=-x4, f3=x2, f4=-x2arrow_forwardShow that the set degree 1 polynomials with natural number coefficients, i.e.{nx+m | n, m∈N}, is countable.arrow_forward
- Describe the Horner algorithm of evaluating a polynomial's value. Why is it better than other methods? (b) Use the Horner algorithm to compute f(−1)f(−1) for f(x)=2x4−x3+3x2−5x−2f(x)=2x4−x3+3x2−5x−2.arrow_forwardFor f(x), g(x), and Zn[x]in the given question ,, find the greatest common divisor d(x) of f(x) and g(x) in Zn[x]. f(x) = x3+ 2x2+ 2, g(x) = 2x5+ 2x4+ x2 + 2, in Z3[x].arrow_forwardFind a formula for the polynomial P(x) with degree 3 leading coefficient 1 zeros at 8, 5, and 0 P(x)=arrow_forward
- Find the Taylor polynomials p3 and p4 centered at a = 0 for ƒ(x) = (1 + x)-3.arrow_forwardFind a polynomial with integer coefficients that satisfies the given conditions. P has degree 3 and zeros 5 and i. P(x)= ?arrow_forwardfind a formula for the polynomial p(x) with degree 3 leading coefficient 1 zeros at 3, -5, and 8arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,