# Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 19

    Exercise for the break

### 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 $X^{m}$?
    Exercises

### Exercise

Compute the polynomial
    $(3X^{2}-5X+4)\cdot (X^{3}-6X^{2}+1)-(4X^{3}+2X^{2}-2X+3)\cdot (-2X^{2}-5X)$
in the polynomial ring $\mathbb {Q} [X]$.

### 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

Show that the polynomial ring $K[X]$ over a field $K$ is a commutative ring.

### 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

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

### 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

Write the polynomial
    $X^{3}+2X^{2}-3X+4$
in the new variable $U=X+2$.

### Exercise

Polynomial/Substitution/iX-3+2i in X^3-2iX^2+4X-1/Exercise

### 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 $T=2X^{2}+3X-1$.
The field $\mathbb {Z} /(7)$ was introduced in example *****.

### Exercise

Division with remainder (polynomials)/Z mod 7/5x^4+3x^3+5x^2+3x+6/3x^2+6x+4/Exercise
In den following exercises, we deal with the solution formula for quadratic equations.

### Exercise

Quadratic equation/R/Solve/1/Exercise

### Exercise

Quadratic equation/R/Completion/2/Exercise

### Exercise

Quadratic equation/R/Solution formula/Fact/Proof/Exercise

### Exercise

Quadratic equation/Side lengths from perimeter and area of a rectangle/Exercise

### Exercise

Quadratic polynomial/Minimum/1/Exercise

### Exercise

Biquadratic equation/R/Solve/1/Exercise

### Exercise

Normed cubic polynomials/Intersection points/1/Exercise

### Exercise

Finite geometric series/Polynomial/Exercise

### Exercise

Polynomial/Complex zeroes of X^3-1/Exercise

### 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

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.

### Exercise

Polynomial ring/Power of X/Factors/Exercise

### Exercise

Polynomial ring/Field/One variable/Product of linear factors/Divisors/Exercise

### Exercise

Let $P\in \mathbb {R} [X]$ be a polynomial of degree $d\geq 1$, $P\neq X$. Show that $P$ has at most $d$ fixed points.

### 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 $d$ be a fixed positive natural number. Show that, for every integer number $n$, there exists a uniquely determined integer number $q$ and a uniquely determined natural number $r$, $0\leq r\leq d-1$, such that
    $n=qd+r\,$
holds.

### 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

Complex polynomial/Surjective/Exercise

### 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

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

### 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

Functions/R/Composition and multiplication/Exercise
    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 (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 Division with remainder (polynomials)/Z mod 7/6X^4+2X^3+4X^2+2X+5/5X^2+3X+2/Exercise

Division with remainder (polynomials)/Z mod 7/6X^4+2X^3+4X^2+2X+5/5X^2+3X+2/Exercise

### Exercise Polynomials/Formula for X^odd+1/Exercise

Polynomials/Formula for X^odd+1/Exercise

### Exercise Normed cubic polynomials/Intersection points/2/Exercise

Normed cubic polynomials/Intersection points/2/Exercise

### 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 $2$.
    The exercise to give up

### Exercise (7 marks)

Two people, $A$ and $B$, play polynomial-guessing. In this game, $A$ imagines a polynomial $P(x)$, where all coefficients are in $\mathbb {N}$. Person $B$ is allowed to ask for the values $P(n_{1}),P(n_{2}),\ldots ,P(n_{r})$ for certain natural numbers $n_{1},n_{2},\ldots ,n_{r}$. Here, $B$ may choose these numbers arbitrarily, taking the previous answers into account. The goal is to find the polynomial.
Describe a strategy for $B$ to find always the polynomial, where the number of questions is (independent of the polynomial) bounded.