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

    Exercises

### Exercise

Calculate in the polynomial ring $\mathbb {C} [X]$ the product
    $((4+{\mathrm {i} })X^{2}-3X+9{\mathrm {i} })\cdot ((-3+7{\mathrm {i} })X^{2}+(2+2{\mathrm {i} })X-1+6{\mathrm {i} }).$

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Prove the following properties concerning the degree of a polynomial:
    $\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,$
    $\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.$

### Exercise

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

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $a\in K$. Prove that the evaluating function
    $\psi \colon K[X]\longrightarrow K,P\longmapsto P(a),$
satisfies the following properties (here let $P,Q\in K[X]$).
    $(P+Q)(a)=P(a)+Q(a)\,.$
    $(P\cdot Q)(a)=P(a)\cdot Q(a)\,.$
    $1(a)=1\,.$

### Exercise

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

### Exercise

Show that
    $z={\sqrt[{3}]{-1+{\sqrt {2}}}}+{\sqrt[{3}]{-1-{\sqrt {2}}}}\,$
is a zero of the polynomial
    $X^{3}+3X+2.$

### Exercise

Evaluate the polynomial
    $2X^{3}-5X^{2}-4X+7$
replacing the variable $X$ by the complex number $2-5{\mathrm {i} }$.

### Exercise

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

### Exercise

Let
    $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,$
denote a real polynomial with $a_{n}>0$. Describe in dependence on the coefficients $a_{0},\ldots ,a_{n}$ a bound $b$ such that
    $P(x)>0\,$
holds for all $x\geq b$.

### Exercise

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

### Exercise

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

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Show that every polynomial $P\in K[X]$, $P\neq 0$, can be decomposed as a product
    $P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,$
where $\mu _{j}\geq 1$ and $Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers $\lambda _{1},\ldots ,\lambda _{k}$ and the exponents $\mu _{1},\ldots ,\mu _{k}$ are uniquely determined apart from the order.
The exponents $\mu _{i}$ are called the order of zero of the zero $\lambda _{i}$ in the polynomial.

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise *

Find a
polynomial
    $f=a+bX+cX^{2}\,$
with $a,b,c\in \mathbb {R}$, such that the following conditions hold.
    $f(-1)=2,\,f(1)=0,\,f(3)=5.$

### Exercise

Find a polynomial
    $f=a+bX+cX^{2}+dX^{3}\,$
with $a,b,c,d\in \mathbb {R}$, such that the following conditions hold.
    $f(0)=1,\,f(1)=2,\,f(2)=0,\,f(-1)=1.$

### Exercise

Let $K$ be an ordered field and let $R=K[X]$ be the
polynomial ring over $K$. Let
    $P={\left\{F\in K[X]\mid {\text{The leading coefficient of }}F{\text{ is positive}}\right\}}\,.$
Show that $P$ fulfils the following three properties.
* Either $F\in P$ or $-F\in P$ or $F=0$.
* If $F,G\in P$, then also $F+G\in P$.
* If $F,G\in P$, then also $F\cdot G\in P$.

### Exercise

Let $K[X]$ be the polynomial ring over a field $K$. Show that the set
    ${\left\{{\frac {P}{Q}}\mid P,Q\in K[X],\,Q\neq 0\right\}}$
with a suitable addition and multiplication is a field, where two fractions ${\frac {P}{Q}}$ and ${\frac {P'}{Q'}}$ are considered to be equal if $PQ'=P'Q$.

### Exercise

Compute in $\mathbb {Q} (X)$ the following expressions.
* The product
    ${\frac {2X^{3}-5X^{2}+X-1}{X^{2}-2X+6}}\cdot {\frac {X^{2}+3}{5X^{3}-4X^{2}-7}}.$
* The sum
    ${\frac {4X^{3}-X^{2}+6X-2}{X^{2}-4X-3}}+{\frac {X^{2}-3}{3X^{2}+5}}.$
* The inverse of
    ${\frac {6X^{3}-9X^{2}+5X-1}{X^{4}-4X^{3}+3X^{2}-8X-3}}.$

### Exercise

Sketch the graph of the following rational functions
    $f=g/h\colon U\longrightarrow \mathbb {R} ,$
where each time $U$ is the complement set of the set of the zeros of the denominator polynomial $h$.
* $1/x$,
* $1/x^{2}$,
* $1/(x^{2}+1)$,
* $x/(x^{2}+1)$,
* $x^{2}/(x^{2}+1)$,
* $x^{3}/(x^{2}+1)$,
* $(x-2)(x+2)(x+4)/(x-1)x(x+1)$.

### Exercise

Let $K$ be an ordered field, let $K[X]$ be the polynomial ring over $K$ and set
    $Q=K(X)\,,$
the field of rational functions over $K$. Show, using Exercise 6.19 , that $Q$ can be made into an ordered field, which is not an archimedean ordered field.

### Exercise

Let $x$ be a real number, $x\neq 1$. Prove for $n\in \mathbb {N}$ by induction the relation
    $\sum _{k=0}^{n}x^{k}={\frac {x^{n+1}-1}{x-1}}\,.$

### Exercise

Compute the compositions $f\circ g$ and $g\circ f$ for the rational functions
    $f(x)={\frac {2x^{2}-4x+3}{x-2}}\,\,{\text{ and }}\,\,g(x)={\frac {x+1}{x^{2}-4}}.$

### Exercise

Show that the composition of rational functions is again a rational function.
    Hand-in-exercises

### Exercise (3 marks)

Compute in the polynomial ring $\mathbb {C} [X]$ the product
    $((4+{\mathrm {i} })X^{3}-{\mathrm {i} }X^{2}+2X+3+2{\mathrm {i} })\cdot ((2-{\mathrm {i} })X^{3}+(3-5{\mathrm {i} })X^{2}+(2+{\mathrm {i} })X+1+5{\mathrm {i} }).$

### Exercise (3 marks)

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

### Exercise (4 marks)

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

### Exercise (2 marks)

Prove the formula
    $X^{u}+1=(X+1){\left(X^{u-1}-X^{u-2}+X^{u-3}-\cdots +X^{2}-X+1\right)}\,$
for $u$ odd.

### Exercise (4 marks)

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

### Exercise (4 marks)

Find a polynomial $f$ of degree $\leq 3$ for which
    $f(0)=-1,\,f(-1)=-3,\,f(1)=7,\,f(2)=21$
holds.