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

    Exercise for the break

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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 $X^{m}$?
    Exercises

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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 ringMDLD/polynomial ring $\mathbb {Q} [X]$.

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Calculate in the polynomial ringMDLD/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 Exercise 19.4
Show that the polynomial ringMDLD/polynomial ring (1 K) $K[X]$ over a fieldMDLD/field $K$ is a commutative ring.MDLD/commutative ring

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

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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 $PQ\neq 0$.

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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 $2-5{\mathrm {i} }$.
===Exercise Exercise 19.8
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 $P,Q\in K[X]$).
    $(P+Q)(a)=P(a)+Q(a)\,.$
    $(P\cdot Q)(a)=P(a)\cdot Q(a)\,.$
    $1(a)=1\,.$

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Write the polynomialMDLD/polynomial (1K)
    $X^{3}+2X^{2}-3X+4$
in the new variable $U=X+2$.

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Let the complex polynomials
    $P=X^{3}-2{\mathrm {i} }X^{2}+4X-1\,\,{\text{ and }}\,\,Q={\mathrm {i} }X-3+2{\mathrm {i} }$
be given. Compute $P(Q)$ (that is, plug in $Q$ into $P$).

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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 3.9 .

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Perform, in the polynomial ring $\mathbb {Z} /(7)[X]$, the division with remainder ${\frac {P}{T}}$, where $P=5X^{4}+3X^{3}+5X^{2}+3X+6$ and $T=3X^{2}+6X+4$.
In den following exercises, we deal with the solution formula for quadratic equations.

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Solve the quadratic equation $3x^{2}-6x+2=0$ over $\mathbb {R}$.

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Solve the real-quadratic equation ${\frac {3}{4}}x^{2}+{\frac {2}{7}}x-{\frac {4}{5}}=0$ by completing the square.

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Prove the solution formula for real quadratic equations.

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Suppose that, of a rectangle, its perimeter $U$ and its area $A$ are known. Determine the lengths of the edges of the rectangle.

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Determine the minimal value of the real function
    $f(x)=x^{2}-3x+{\frac {4}{3}}\,.$

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Solve the biquadratic equation $x^{4}+7x^{2}-11=0$ over $\mathbb {R}$.

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Determine the $x$-coordinates of the intersection points of the graphs of the two real polynomials
    $P=X^{3}+4X^{2}-7X+1\,$
and
    $Q=X^{3}-2X^{2}+5X+3\,.$

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Prove the formula
    $X^{n}-1=(X-1)(X^{n-1}+X^{n-2}+X^{n-3}+\cdots +X^{2}+X+1)\,.$

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Determine all complex zeroes of the polynomial
    $X^{3}-1,$
and describe the factorization of this polynomial in $\mathbb {R} [X]$ and in $\mathbb {C} [X]$.

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

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

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Let $K$ be a field,MDLD/field and let $K[X]$ be the polynomial ringMDLD/polynomial ring over $K$. Let $P=X^{n}\in K[X]$ with $n\geq 1$. Show that all normed divisors of $P$ have the form $X^{k}$, $1\leq k\leq n$.

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Let $K$ be a field,MDLD/field and let $K[X]$ be the polynomial ringMDLD/polynomial ring over $K$, and let $P\in K[X]$ be a polynomial which has a factorization into linear factors. Let $T$ be a divisorMDLD/divisor (polynomial) of $P$. Show that $T$ has also a factorization into linear factors, and that the multiplicity of every linear factor $X-a$ in $T$ is bounded from above by its multiplicity in $P$.

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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.MDLD/fixed points

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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 $d$ 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.

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Let $F\in \mathbb {C} [X]$ 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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Let $P\in \mathbb {C} [X]$ denote a nonconstant polynomial. Show that the mapping
    $\mathbb {C} \longrightarrow \mathbb {C} ,z\longmapsto P(z),$
is surjective.

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Let $K[X]$ be the polynomial ringMDLD/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 $PQ'=P'Q$.

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

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

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Let
    $f,g,h\colon \mathbb {R} \longrightarrow \mathbb {R}$
denote functions.
a) Show the equality
    ${\left(h\cdot g\right)}\circ f={\left(h\circ f\right)}\cdot {\left(g\circ f\right)}\,.$
b) Show by an example that the equality
    ${\left(h\circ g\right)}\cdot f={\left(h\cdot f\right)}\circ {\left(g\cdot f\right)}\,$
does not hold in general.
    Hand-in-exercises

### Exercise (3 marks) Create referencenumber

Compute in the polynomial ringMDLD/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) Create referencenumber

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 (3 marks) Create referencenumber

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

### Exercise (2 marks) Create referencenumber

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 (3 marks) Create referencenumber

Determine the $x$-coordinates of the intersection points of the graphs of the two real polynomials
    $P=X^{3}+8X^{2}-8X+3\,$
and
    $Q=X^{3}-5X^{2}+4X+7\,.$

### Exercise (4 marks) Create referencenumber

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) Create referencenumber

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.