- Exercises
Calculate in the
polynomial ringMDLD/polynomial ring
the product
-
Let
be a field and let
be the polynomial ring over
. Prove the following properties concerning the
degreeMDLD/degree
of a polynomial:
-
![{\displaystyle {}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a8ce0404f58829f08ff4d244c72c3fe796562a)
-
![{\displaystyle {}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6155f05f6c52942d5fba8639d141e60f03d1a88f)
Show that in a
polynomial ringMDLD/polynomial ring (1K)
over a
fieldMDLD/field
, the following statement holds: if
are not zero, then also
.
Let
be a field and let
be the polynomial ring over
. Let
.
Prove that the evaluating function
-
satisfies the following properties
(here let
).
-
![{\displaystyle {}(P+Q)(a)=P(a)+Q(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fad78a123e99e89811832de92a9ce0420d63bc6)
-
![{\displaystyle {}(P\cdot Q)(a)=P(a)\cdot Q(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da262e38a71464df7ce010f7450cd21972a9cdd2)
-
![{\displaystyle {}1(a)=1\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8906134ad3e345aa4335f32a84b45659f959435a)
Insert into the polynomial
the number
.
Show that
-
![{\displaystyle {}z={\sqrt[{3}]{-1+{\sqrt {2}}}}+{\sqrt[{3}]{-1-{\sqrt {2}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a39d977536d0c3a9667948a3622f46787086352)
is a zero of the polynomial
-
Evaluate the
polynomialMDLD/polynomial (1K)
-
replacing the variable
by the
complex numberMDLD/complex number
.
Show that the composition
(the inserting of a polynomial into another one)
of two polynomials is again a polynomial.
Let
-
![{\displaystyle {}P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0773d24c76f3393ff661a9b8cc70d49f6cbe970f)
denote a real polynomial with
.
Describe in dependence on the coefficients
a bound
such that
-
![{\displaystyle {}P(x)>0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03fd2ca291f2d7f95639d96f56af20cbd4578df3)
holds for all
.
Let
be a
fieldMDLD/field
and let
be the
polynomial ringMDLD/polynomial ring
over
. What is the result when we divide
(with remainder)
a polynomial
by
?
Perform, in the polynomial ring
, the division with remainder
, where
,
and
.
Let
be a field and let
be the polynomial ring over
. Show that every polynomial
,
,
can be decomposed as a product
-
![{\displaystyle {}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb27f1dab00668431a5b7bc0d34960909e92fa7)
where
and
is a polynomial with no roots (no zeroes). Moreover, the different numbers
and the exponents
are uniquely determined apart from the order.
The exponents
are called the order of zero of the zero
in the polynomial.
Let
and
denote different
normed polynomialsMDLD/normed polynomials
of degree
over a field
. How many intersection points may both graphs have at most?
Let
be a
non-constantMDLD/non-constant (map)
polynomial.MDLD/polynomial (1K)
Prove that
can be decomposed as a product of
linear factors.MDLD/linear factors (1K)
Determine the smallest real number for which the
Bernoulli inequality
with exponent
holds.
Let
be a
polynomialMDLD/polynomial (1K)
with
realMDLD/real
coefficients and let
be a
rootMDLD/root
of
. Show that also the
complex conjugateMDLD/complex conjugate
is a root of
.
Find a
polynomialMDLD/polynomial (1K)
-
![{\displaystyle {}f=a+bX+cX^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8049b64104867c582e007106c9abfb54ea2ce680)
with
,
such that the following conditions hold.
-
Find a
polynomialMDLD/polynomial (1K)
-
![{\displaystyle {}f=a+bX+cX^{2}+dX^{3}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15cb632f45ca47ff084ba7da6076d1323b9a529)
with
,
such that the following conditions hold.
-
===Exercise Exercise 6.19
change===
Let
be an
ordered fieldMDLD/ordered field
and let
be the
polynomial ringMDLD/polynomial ring
over
. Let
-
![{\displaystyle {}P={\left\{F\in K[X]\mid {\text{The leading coefficient of }}F{\text{ is positive}}\right\}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28af2d8d125881eb37ab39c641c134197f85a5c)
Show that
fulfils the following three properties.
- Either
or
or
.
- If
,
then also
.
- If
,
then also
.
Let
be the
polynomial ringMDLD/polynomial ring
over a field
. Show that the set
-
with a suitable addition and multiplication is a field, where two fractions
and
are considered to be equal if
.
Compute in
the following expressions.
- The product
-
- The sum
-
- The inverse of
-
Sketch the graph of the following
rational functionsMDLD/rational functions (K)
-
where each time
is the
complement setMDLD/complement set
of the set of the zeros of the denominator polynomial
.
,
,
,
,
,
,
.
Let
be an
ordered field,MDLD/ordered field
let
be the
polynomial ringMDLD/polynomial ring
over
and set
-
![{\displaystyle {}Q=K(X)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5352ef40e80bdf7b65cf4aa4653065c07498810b)
the
field of rational functionsMDLD/field of rational functions
over
. Show, using
Exercise 6.19
,
that
can be made into an ordered field, which is not an
archimedean ordered field.MDLD/archimedean ordered field
Let
be a
real number,MDLD/real number
.
Prove for
by induction the relation
-
![{\displaystyle {}\sum _{k=0}^{n}x^{k}={\frac {x^{n+1}-1}{x-1}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/645a9890a9d4cc3af8b69680209db556d1f3997f)
Compute the
compositionsMDLD/compositions
and
for the
rational functionsMDLD/rational functions (K)
-
Show that the
compositionMDLD/composition
of
rational functionsMDLD/rational functions (K)
is again a rational function.
- Hand-in-exercises
Compute in the
polynomial ringMDLD/polynomial ring
the product
-
Perform in the polynomial ring
the division with remainder
, where
and
.
Perform, in the polynomial ring
the division with remainder
, where
-
![{\displaystyle {}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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86955b125076af16afe619848e82fd1715f0e3c4)
and
-
![{\displaystyle {}T=X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddfc63f197ba5f95144f666cccb5d48b0c1cadfd)
Prove the formula
-
![{\displaystyle {}X^{u}+1=(X+1){\left(X^{u-1}-X^{u-2}+X^{u-3}-\cdots +X^{2}-X+1\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0952c155ede42e5fd084ed7988ac211fd7454c3e)
for
odd.
Let
be a non-constant polynomial with real coefficients. Prove that
can be written as a product of real polynomials of degrees
or
.
Find a
polynomialMDLD/polynomial (1K)
of degree
for which
-
holds.