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

Exercises

### Exercise

Calculate in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the product

${\displaystyle ((4+{\mathrm {i} })X^{2}-3X+9{\mathrm {i} })\cdot ((-3+7{\mathrm {i} })X^{2}+(2+2{\mathrm {i} })X-1+6{\mathrm {i} }).}$

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Prove the following properties concerning the degree of a polynomial:

1. ${\displaystyle {}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,}$
2. ${\displaystyle {}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.}$

### Exercise

Show that in a polynomial ring over a field ${\displaystyle {}K}$, the following statement holds: if ${\displaystyle {}P,Q\in K[X]}$ are not zero, then also ${\displaystyle {}PQ\neq 0}$.

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Let ${\displaystyle {}a\in K}$. Prove that the evaluating function

${\displaystyle \psi \colon K[X]\longrightarrow K,P\longmapsto P(a),}$

satisfies the following properties (here let ${\displaystyle {}P,Q\in K[X]}$).

1. ${\displaystyle {}(P+Q)(a)=P(a)+Q(a)\,.}$
2. ${\displaystyle {}(P\cdot Q)(a)=P(a)\cdot Q(a)\,.}$
3. ${\displaystyle {}1(a)=1\,.}$

### Exercise

Insert into the polynomial ${\displaystyle {}2X^{4}+X^{3}-3X^{2}+X+5}$ the number ${\displaystyle {}{\sqrt {2}}}$.

### Exercise

Show that

${\displaystyle {}z={\sqrt[{3}]{-1+{\sqrt {2}}}}+{\sqrt[{3}]{-1-{\sqrt {2}}}}\,}$

is a zero of the polynomial

${\displaystyle X^{3}+3X+2.}$

### Exercise

Evaluate the polynomial

${\displaystyle 2X^{3}-5X^{2}-4X+7}$

replacing the variable ${\displaystyle {}X}$ by the complex number ${\displaystyle {}2-5{\mathrm {i} }}$.

### Exercise

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

### Exercise

Let

${\displaystyle {}P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,}$

denote a real polynomial with ${\displaystyle {}a_{n}>0}$. Describe in dependence on the coefficients ${\displaystyle {}a_{0},\ldots ,a_{n}}$ a bound ${\displaystyle {}b}$ such that

${\displaystyle {}P(x)>0\,}$

holds for all ${\displaystyle {}x\geq b}$.

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. What is the result when we divide (with remainder) a polynomial ${\displaystyle {}P}$ by ${\displaystyle {}X^{m}}$?

### Exercise

Perform, in the polynomial ring ${\displaystyle {}\mathbb {Q} [X]}$, the division with remainder ${\displaystyle {}{\frac {P}{T}}}$, where ${\displaystyle {}P=3X^{4}+7X^{2}-2X+5}$, and ${\displaystyle {}T=2X^{2}+3X-1}$.

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Show that every polynomial ${\displaystyle {}P\in K[X]}$, ${\displaystyle {}P\neq 0}$, can be decomposed as a product

${\displaystyle {}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,}$

where ${\displaystyle {}\mu _{j}\geq 1}$ and ${\displaystyle {}Q}$ is a polynomial with no roots (no zeroes). Moreover, the different numbers ${\displaystyle {}\lambda _{1},\ldots ,\lambda _{k}}$ and the exponents ${\displaystyle {}\mu _{1},\ldots ,\mu _{k}}$ are uniquely determined apart from the order.

The exponents ${\displaystyle {}\mu _{i}}$ are called the order of zero of the zero ${\displaystyle {}\lambda _{i}}$ in the polynomial.

### Exercise

Let ${\displaystyle {}P}$ and ${\displaystyle {}Q}$ denote different normed polynomials of degree ${\displaystyle {}d}$ over a field ${\displaystyle {}K}$. How many intersection points may both graphs have at most?

### Exercise

Let ${\displaystyle {}F\in \mathbb {C} [X]}$ be a non-constant polynomial. Prove that ${\displaystyle {}F}$ can be decomposed as a product of linear factors.

### Exercise

Determine the smallest real number for which the Bernoulli inequality with exponent ${\displaystyle {}n=3}$ holds.

### Exercise

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a polynomial with real coefficients and let ${\displaystyle {}z\in \mathbb {C} }$ be a root of ${\displaystyle {}P}$. Show that also the complex conjugate ${\displaystyle {}{\overline {z}}}$ is a root of ${\displaystyle {}P}$.

### Exercise *

Find a polynomial

${\displaystyle {}f=a+bX+cX^{2}\,}$

with ${\displaystyle {}a,b,c\in \mathbb {R} }$, such that the following conditions hold.

${\displaystyle f(-1)=2,\,f(1)=0,\,f(3)=5.}$

### Exercise

Find a polynomial

${\displaystyle {}f=a+bX+cX^{2}+dX^{3}\,}$

with ${\displaystyle {}a,b,c,d\in \mathbb {R} }$, such that the following conditions hold.

${\displaystyle f(0)=1,\,f(1)=2,\,f(2)=0,\,f(-1)=1.}$

### Exercise

Let ${\displaystyle {}K}$ be an ordered field and let ${\displaystyle {}R=K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Let

${\displaystyle {}P={\left\{F\in K[X]\mid {\text{The leading coefficient of }}F{\text{ is positive}}\right\}}\,.}$

Show that ${\displaystyle {}P}$ fulfils the following three properties.

1. Either ${\displaystyle {}F\in P}$ or ${\displaystyle {}-F\in P}$ or ${\displaystyle {}F=0}$.
2. If ${\displaystyle {}F,G\in P}$, then also ${\displaystyle {}F+G\in P}$.
3. If ${\displaystyle {}F,G\in P}$, then also ${\displaystyle {}F\cdot G\in P}$.

### Exercise

Let ${\displaystyle {}K[X]}$ be the polynomial ring over a field ${\displaystyle {}K}$. Show that the set

${\displaystyle {\left\{{\frac {P}{Q}}\mid P,Q\in K[X],\,Q\neq 0\right\}}}$

with a suitable addition and multiplication is a field, where two fractions ${\displaystyle {}{\frac {P}{Q}}}$ and ${\displaystyle {}{\frac {P'}{Q'}}}$ are considered to be equal if ${\displaystyle {}PQ'=P'Q}$.

### Exercise

Compute in ${\displaystyle {}\mathbb {Q} (X)}$ the following expressions.

1. The product
${\displaystyle {\frac {2X^{3}-5X^{2}+X-1}{X^{2}-2X+6}}\cdot {\frac {X^{2}+3}{5X^{3}-4X^{2}-7}}.}$
2. The sum
${\displaystyle {\frac {4X^{3}-X^{2}+6X-2}{X^{2}-4X-3}}+{\frac {X^{2}-3}{3X^{2}+5}}.}$
3. The inverse of
${\displaystyle {\frac {6X^{3}-9X^{2}+5X-1}{X^{4}-4X^{3}+3X^{2}-8X-3}}.}$

### Exercise

Sketch the graph of the following rational functions

${\displaystyle f=g/h\colon U\longrightarrow \mathbb {R} ,}$

where each time ${\displaystyle {}U}$ is the complement set of the set of the zeros of the denominator polynomial ${\displaystyle {}h}$.

1. ${\displaystyle {}1/x}$,
2. ${\displaystyle {}1/x^{2}}$,
3. ${\displaystyle {}1/(x^{2}+1)}$,
4. ${\displaystyle {}x/(x^{2}+1)}$,
5. ${\displaystyle {}x^{2}/(x^{2}+1)}$,
6. ${\displaystyle {}x^{3}/(x^{2}+1)}$,
7. ${\displaystyle {}(x-2)(x+2)(x+4)/(x-1)x(x+1)}$.

### Exercise

Let ${\displaystyle {}K}$ be an ordered field, let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$ and set

${\displaystyle {}Q=K(X)\,,}$

the field of rational functions over ${\displaystyle {}K}$. Show, using Exercise 6.19 , that ${\displaystyle {}Q}$ can be made into an ordered field, which is not an archimedean ordered field.

### Exercise

Let ${\displaystyle {}x}$ be a real number, ${\displaystyle {}x\neq 1}$. Prove for ${\displaystyle {}n\in \mathbb {N} }$ by induction the relation

${\displaystyle {}\sum _{k=0}^{n}x^{k}={\frac {x^{n+1}-1}{x-1}}\,.}$

### Exercise

Compute the compositions ${\displaystyle {}f\circ g}$ and ${\displaystyle {}g\circ f}$ for the rational functions

${\displaystyle f(x)={\frac {2x^{2}-4x+3}{x-2}}\,\,{\text{ and }}\,\,g(x)={\frac {x+1}{x^{2}-4}}.}$

### Exercise

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

Hand-in-exercises

### Exercise (3 marks)

Compute in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the product

${\displaystyle ((4+{\mathrm {i} })X^{3}-{\mathrm {i} }X^{2}+2X+3+2{\mathrm {i} })\cdot ((2-{\mathrm {i} })X^{3}+(3-5{\mathrm {i} })X^{2}+(2+{\mathrm {i} })X+1+5{\mathrm {i} }).}$

### Exercise (3 marks)

Perform in the polynomial ring ${\displaystyle {}\mathbb {Q} [X]}$ the division with remainder ${\displaystyle {}{\frac {P}{T}}}$, where ${\displaystyle {}P=5X^{4}-6X^{3}+{\frac {3}{5}}X^{2}-{\frac {1}{2}}X+5}$ and ${\displaystyle {}T={\frac {1}{7}}X^{2}+{\frac {3}{7}}X-1}$.

### Exercise (4 marks)

Perform, in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the division with remainder ${\displaystyle {}{\frac {P}{T}}}$, where

${\displaystyle {}P=(5+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X^{4}+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }X^{2}+(3-2X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X-1\,}$

and

${\displaystyle {}T=X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }\,.}$

### Exercise (2 marks)

Prove the formula

${\displaystyle {}X^{u}+1=(X+1){\left(X^{u-1}-X^{u-2}+X^{u-3}-\cdots +X^{2}-X+1\right)}\,}$

for ${\displaystyle {}u}$ odd.

### Exercise (4 marks)

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a non-constant polynomial with real coefficients. Prove that ${\displaystyle {}P}$ can be written as a product of real polynomials of degrees ${\displaystyle {}1}$ or ${\displaystyle {}2}$.

### Exercise (4 marks)

Find a polynomial ${\displaystyle {}f}$ of degree ${\displaystyle {}\leq 3}$ for which

${\displaystyle f(0)=-1,\,f(-1)=-3,\,f(1)=7,\,f(2)=21}$

holds.