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

*Exercises*

Calculate in the polynomial ring the product

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

Show that in a polynomial ring over a 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 polynomial

replacing the variable by the 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 and let be the 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 polynomials of degree over a field . How many intersection points may both graphs have at most?

Let be a non-constant polynomial. Prove that can be decomposed as a product of linear factors.

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

Let be a polynomial with real coefficients and let be a root of . Show that also the complex conjugate is a root of .

Find a polynomial

with , such that the following conditions hold.

Find a polynomial

with , such that the following conditions hold.

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

Show that fulfils the following three properties.

- Either or or .
- If , then also .
- If , then also .

Let be the 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.

- The product
- The sum
- The inverse of

Sketch the graph of the following rational functions

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

- ,
- ,
- ,
- ,
- ,
- ,
- .

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

the
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.

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

Compute the compositions and for the rational functions

Show that the composition of rational functions is again a rational function.

*Hand-in-exercises*

### Exercise (3 marks)

Compute in the polynomial ring the product

### Exercise (3 marks)

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

### Exercise (4 marks)

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

and

### Exercise (2 marks)

Prove the formula

for odd.

### Exercise (4 marks)

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

### Exercise (4 marks)

Find a polynomial of degree for which

holds.

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