Lab 4

Exercise 1: Polynomial Properties

  1. Define the polynomial ( p(x) = 6x^4 - 5x^3 + 4x^2 - 3x + 2 ).
  2. Compute and display:
    • The degree and leading coefficient of ( p(x) ).
    • The list of coefficients and monomials.
    • The sum of all coefficients.
  3. Determine whether ( p(x) ) is monic (has a leading coefficient of 1). If not, normalize it to make it monic.

Exercise 2: Formal power series

Let R be a commutative ring with 1. Define the set R[[x]] of formal power series in indeterminate x to be all formal infinite sums
\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x+…
Define addition and multiplication on R[[x]] in the same way as for power series with real or complex coefficients.

  1. Prove that R[ ] is a commutative ring with 1.
  2. Show that 1-x is a unit in R[ ].
  3. Prove that \sum_{n=0}^\infty a_n x^n is a unit in R[ ] if and only if a_0 is a unit in R.
  4. Prove that if R is an integral domain, then so is R[ ].

Exercise 3: Division and Modular Arithmetic

  1. Define the polynomial ( p(x) = 7x^4 + 3x^3 - 5x^2 + x - 8 ).
  2. Divide ( p(x) ) by ( q(x) = 2x^2 - 1 ) over:
    • The integers (( \mathbb{Z} )).
    • The rationals (( \mathbb{Q} )).
    • The finite field ( \mathbb{Z}_5 ).
  3. Analyze the differences in the results and interpret why they occur.

Exercise 4: Factoring

  1. Define the polynomial ( p(x) = x^4 + 2x^3 + x^2 - x - 2 ).
  2. Factor ( p(x) ) over:
    • The integers (( \mathbb{Z} )).
    • The rationals (( \mathbb{Q} )).
    • The complex numbers (( \mathbb{C} )).
    • The finite field ( \mathbb{Z}_7 ).
  3. Verify that the product of the factors in each case reconstructs ( p(x) ).

Exercise 5: Multivariate Polynomial

  1. Define ( f(x, y) = 4x^3y^2 - 2x^2y + 3xy^3 - 5y^2 + 7 ) and ( g(x, y) = x^2y - xy^2 + 2 ).
  2. Perform the following:
    • Compute the GCD of ( f(x, y) ) and ( g(x, y) ).
    • Divide ( f(x, y) ) by ( g(x, y) ) and verify that ( f(x, y) = g(x, y)q(x, y) + r(x, y) ).
    • Integrate ( f(x, y) ) with respect to ( x ), treating ( y ) as a constant.

Exercise 6: Rational and Partial Fraction Decomposition

  1. Define the rational polynomial: [ r(x) = \frac{x^4 + 3x^3 - 2x^2 + x - 1}{x^3 - x + 2} ]
  2. Perform the following:
    • Simplify ( r(x) ) and express it in terms of polynomial and proper fraction parts.
    • Decompose ( r(x) ) into partial fractions.
  3. Redefine ( r(x) ) with ( q(x) = x^3 - x + 2 ) factored over ( \mathbb{C} ) and compute the new partial fraction decomposition. Interpret the role of the roots of ( q(x) ) in the result.