Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 6/refcontrol



Exercises

Calculate in the polynomial ringMDLD/polynomial ring the product


Let be a field and let be the polynomial ring over . Prove the following properties concerning the degreeMDLD/degree (polynomial) of a polynomial:


Show that in a polynomial ringMDLD/polynomial ring (1K) over a fieldMDLD/field , the following statement holds: if are not zero, then also .


Let be a field and let be the polynomial ring over . Let . Prove that the evaluating function

satisfies the following properties (here let ).


Insert into the polynomial the number .


Show that

is a zero of the polynomial


Evaluate the polynomialMDLD/polynomial (1K)

replacing the variable by the complex numberMDLD/complex number .


Show that the composition (the inserting of a polynomial into another one) of two polynomials is again a polynomial.


Let

denote a real polynomial with . Describe in dependence on the coefficients a bound such that

holds for all .


Let be a field,MDLD/field and let be the polynomial ringMDLD/polynomial ring over . What is the result when we divide (with remainder) a polynomial by ?


Perform, in the polynomial ring , the division with remainder , where , and .


Let be a field and let be the polynomial ring over . Show that every polynomial , , can be decomposed as a product

where and is a polynomial with no roots (no zeroes). Moreover, the different numbers and the exponents are uniquely determined apart from the order.


The exponents are called the order of zero of the zero in the polynomial.

Let and denote different normed polynomialsMDLD/normed polynomials of degree over a field . How many intersection points may both graphs have at most?


Let be a non-constantMDLD/non-constant (map) polynomial.MDLD/polynomial (1K) Prove that can be decomposed as a product of linear factors.MDLD/linear factors (1K)


Determine the smallest real number for which the Bernoulli inequality with exponent holds.


Let be a polynomialMDLD/polynomial (1K) with realMDLD/real coefficients and let be a rootMDLD/root of . Show that also the complex conjugateMDLD/complex conjugate is a root of .


Find a polynomialMDLD/polynomial (1K)

with , such that the following conditions hold.


Find a polynomialMDLD/polynomial (1K)

with , such that the following conditions hold.


===Exercise Exercise 6.19

change===

Let be an ordered fieldMDLD/ordered field and let be the polynomial ringMDLD/polynomial ring over . Let

Show that fulfils the following three properties.

  1. Either or or .
  2. If , then also .
  3. If , then also .


Let be the polynomial ringMDLD/polynomial ring over a field . Show that the set

with a suitable addition and multiplication is a field, where two fractions and are considered to be equal if .


Compute in the following expressions.

  1. The product
  2. The sum
  3. The inverse of


Sketch the graph of the following rational functionsMDLD/rational functions (K)

where each time is the complement setMDLD/complement set of the set of the zeros of the denominator polynomial .

  1. ,
  2. ,
  3. ,
  4. ,
  5. ,
  6. ,
  7. .


Let be an ordered field,MDLD/ordered field let be the polynomial ringMDLD/polynomial ring over and set

the field of rational functionsMDLD/field of rational functions over . Show, using Exercise 6.19 , that can be made into an ordered field, which is not an archimedean ordered field.MDLD/archimedean ordered field


Let be a real number,MDLD/real number . Prove for by induction the relation


Compute the compositionsMDLD/compositions and for the rational functionsMDLD/rational functions (K)


Show that the compositionMDLD/composition of rational functionsMDLD/rational functions (K) is again a rational function.




Hand-in-exercises

Compute in the polynomial ringMDLD/polynomial ring the product


Perform in the polynomial ring the division with remainder , where and .


Perform, in the polynomial ring , the division with remainder , where

and


Prove the formula

for odd.


Let be a non-constant polynomial with real coefficients. Prove that can be written as a product of real polynomials of degrees or .


Find a polynomialMDLD/polynomial (1K) of degree for which

holds.



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