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:

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

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

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

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

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

Exercise

Write the polynomial

$X^{3}+2X^{2}-3X+4$

in the new variable ${}U=X+2$.

Exercise

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 ${}P$).

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 3.9 .

Exercise

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.

Exercise

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

Exercise

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

Exercise

Prove the solution formula for real quadratic equations.

Exercise

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

Exercise

Determine the minimal value of the real function

${}f(x)=x^{2}-3x+{\frac {4}{3}}\,.$

Exercise

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

Exercise

Determine the ${}x$-coordinates of the intersection points of the graphs of the two real polynomials

${}P=X^{3}+4X^{2}-7X+1\,$

and

${}Q=X^{3}-2X^{2}+5X+3\,.$

Exercise

Prove the formula

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

Exercise

Determine all complex zeroes of the polynomial

$X^{3}-1,$

and describe the factorization of this polynomial in ${}\mathbb {R} [X]$ and in ${}\mathbb {C} [X]$.

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

Let ${}K$ be a field, and let ${}K[X]$ be the 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 ${}X^{k}$, ${}1\leq k\leq n$.

Exercise

Let ${}K$ be a field, and let ${}K[X]$ be the polynomial ring over ${}K$, and let ${}P\in K[X]$ be a polynomial that has a factorization into linear factors. Let ${}T$ be a divisor 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 ${}P$.

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

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.

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

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)

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 (3 marks)

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)

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)

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)

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.

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