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:

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

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

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 ${}P,Q\in K[X]$).

1. ${}(P+Q)(a)=P(a)+Q(a)\,.$

2. ${}(P\cdot Q)(a)=P(a)\cdot Q(a)\,.$

3. ${}1(a)=1\,.$

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.

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

1. The product

   ${\frac {2X^{3}-5X^{2}+X-1}{X^{2}-2X+6}}\cdot {\frac {X^{2}+3}{5X^{3}-4X^{2}-7}}.$

2. The sum

   ${\frac {4X^{3}-X^{2}+6X-2}{X^{2}-4X-3}}+{\frac {X^{2}-3}{3X^{2}+5}}.$

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

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.

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