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

Exercises

Exercise Create referencenumber

Calculate in the polynomial ringMDLD/polynomial ring $ℂ[X]$ 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 Create referencenumber

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

1. $\mathrm{deg}(P+Q)\le max\{\mathrm{deg}(P),\mathrm{deg}(Q)\},$

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

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Show that in a polynomial ringMDLD/polynomial ring (1K) over a fieldMDLD/field $K$, the following statement holds: if $P,Q\in K[X]$ are not zero, then also $PQ\ne 0$.

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

$\psi :K[X]⟶K,P⟼P(a),$

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. $1(a)=1.$

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

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Show that

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

is a zero of the polynomial

$X^3+3X+2.$

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Evaluate the polynomialMDLD/polynomial (1K)

$2X^3-5X^2-4X+7$

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

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

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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 $a_0,\dots ,a_n$ a bound $b$ such that

$P(x)>0$

holds for all $x\ge b$.

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

Exercise Create referencenumber

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

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

$P=(X-{\lambda }_1{)}^{{\mu }_1}\cdots (X-{\lambda }_k{)}^{{\mu }_k}\cdot Q$

where ${\mu }_j\ge 1$ and $Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers ${\lambda }_1,\dots ,{\lambda }_k$ and the exponents ${\mu }_1,\dots ,{\mu }_k$ 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.

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

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Let $F\in ℂ[X]$ be a non-constantMDLD/non-constant (map) polynomial.MDLD/polynomial (1K) Prove that $F$ can be decomposed as a product of linear factors.MDLD/linear factors (1K)

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

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Let $P\in ℝ[X]$ be a polynomialMDLD/polynomial (1K) with realMDLD/real coefficients and let $z\in ℂ$ be a rootMDLD/root of $P$. Show that also the complex conjugateMDLD/complex conjugate $\bar{z}$ is a root of $P$.

Exercise * Create referencenumber

Find a polynomialMDLD/polynomial (1K)

$f=a+bX+c{X}^2$

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

$f(-1)=2,f(1)=0,f(3)=5.$

Exercise Create referencenumber

Find a polynomialMDLD/polynomial (1K)

$f=a+bX+c{X}^2+d{X}^3$

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

$f(0)=1,f(1)=2,f(2)=0,f(-1)=1.$

===Exercise Exercise 6.19

change===

Let $K$ be an ordered fieldMDLD/ordered field and let $R=K[X]$ be the polynomial ringMDLD/polynomial ring over $K$. Let

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

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 Create referencenumber

Let $K[X]$ be the polynomial ringMDLD/polynomial ring over a field $K$. Show that the set

$\{\frac{P}{Q}\mid P,Q\in K[X],Q\ne 0\}$

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

Exercise Create referencenumber

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 Create referencenumber

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

$f=g/h:U⟶ℝ,$

where each time $U$ is the complement setMDLD/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 Create referencenumber

Let $K$ be an ordered field,MDLD/ordered field let $K[X]$ be the polynomial ringMDLD/polynomial ring over $K$ and set

$Q=K(X),$

the field of rational functionsMDLD/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.MDLD/archimedean ordered field

Exercise Create referencenumber

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

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

Exercise Create referencenumber

Compute the compositionsMDLD/compositions $f\circ g$ and $g\circ f$ for the rational functionsMDLD/rational functions (K)

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

Exercise Create referencenumber

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

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Compute in the polynomial ringMDLD/polynomial ring $ℂ[X]$ 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) Create referencenumber

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) Create referencenumber

Perform, in the polynomial ring $ℂ[X]$, 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) Create referencenumber

Prove the formula

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

for $u$ odd.

Exercise (4 marks) Create referencenumber

Let $P\in ℝ[X]$ 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) Create referencenumber

Find a polynomialMDLD/polynomial (1K) $f$ of degree $\le 3$ for which

$f(0)=-1,f(-1)=-3,f(1)=7,f(2)=21$

holds.

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