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- •26. Groups. Theorems on uniqueness of the unit and the inverse element; order of an element.
- •27. Groups: subgroups, left and right cosets, factor-groups.
- •28. Polynomials: degree, the greatest common divisor, relatively prime polynomials.
- •29. Polynomials: the Euclid algorithm (the algorithm of sequential dividing).
- •30. Roots of polynomials. Horner’s method. Basic theorem and its consequences
- •Horner’s method
- •Basic theorem and its consequences
28. Polynomials: degree, the greatest common divisor, relatively prime polynomials.
A
common form of an equation of the
th
degree (where
is some positive number) is
(1)
The coefficients
of the equation are arbitrary complex numbers, and the leading
coefficient
is not equal to zero. If an equation (1) is given, it is supposed
always to solve it. However we replace the task of solving (1) by
more common task of studying the left part of the equation
(2), named apolynomial of the
th
degree of
.
For example, the following expressions containing
with negative or fractional degrees:
or
are not polynomials.
For simplified recording polynomials we use the
following symbols:
and etc. Observe that sometimes instead of recording a polynomial in
(2), i.e.by decreasing degrees of
we use a recordby increasing degrees of
.
Two polynomials
and
areequal
(or identically equal)
if their coefficients at the same degrees of unknown are equal.
There are obviously polynomials of the
th
degree for every natural number
.Polynomials of zero degree
are non-zero complex numbers. The number zero is also a polynomial,
but it is a unique polynomial of which the degree is not defined.
If polynomials
and
with complex coefficients written for convenience by increasing
degrees of
are given:
and if for example
,
theirsum
is the polynomial
of which the coefficients are obtained by adding
the coefficients of polynomials
and
at the same degrees of
,
i.e.
.
Let non-zero polynomials
and
with complex coefficients be given. If the remainder of dividing
on
is equal to zero, i.e.
is divided on
without any remainder, then the polynomial
is called adivisor of
the polynomial
.
For example, the polynomial
is a divisor of the polynomial
,
and
is a divisor of
.
A polynomial
is acommon divisor
for
and
if it is a divisor for each of these polynomials. Obviously, all the
polynomials of zero degree are common divisors of
and
.
If there are no other divisors for them, they are calledrelatively
prime. For example, the polynomials
and
are relatively prime.
The greatest common
divisor (gcd) of non-zero polynomials
and
is called such a polynomial
which is their common divisor and it is divided on any other common
divisor of these polynomials. The greatest common divisor of
and
is denoted by
.
For example, the polynomial
is the greatest common divisor of the polynomials
and
.
29. Polynomials: the Euclid algorithm (the algorithm of sequential dividing).
Let
and
be polynomials. Divide
on
and we obtain some remainder
in general. Then divide
on
and we obtain a remainder
,
divide
on
and etc. Since the degrees of the remainders are decreasing, in this
chain of sequential divisions we must reach such a situation when
dividing will be done without remainders and therefore the process is
stopped. That remainder
that divides the preceding remainder
will be the greatest common divisor of the polynomials
and
.
Write the above mentioned in the following form:
The last equality shows that
is a divisor for
.
This implies that both summands of the right part of the penultimate
equality are divided on
,
and therefore
will be a divisor for
.
Further rising up by the same way we obtain that
is a divisor for
,
…,
,
.
Then by the second equality
is a divisor for
,
and therefore by the first equality
is a divisor for
.
Thus,
is a common divisor for
and
.
Take now an arbitrary common divisor
of the polynomials
and
.
Since the left part and the first summand of the right part of the
first equality are divided on
,
is also divided on
.
Passing to the second and next equalities, we obtain by the same way
that
is a divisor of
,
,
… At last if it will be proved that
and
are divide on
then from the penultimate equality we obtain that
is divided on
.
Thus,
will be gcd for
and
.
If
is gcd for
and
then we can choose as gcd the polynomial
where
is an arbitrary non-zero number. In other words, the greatest common
divisor of two polynomials has been determined up to a multiplier of
zero degree. Therefore we assume that the leading coefficient of gcd
pf two polynomials will be equal to 1.
Using this convention, we can say that two polynomials are relatively prime iff their gcd is equal to 1.
Theorem 1. If
is the greatest common divisor of
and
then it can be found such polynomials
and
that
(2)
We can assume that if the degrees of
and
is greater than zero then the degree of
is less than the degree of
,
and the degree of
is less than the degree of
.
Proof: The proof
is based on the equalities (1). If we take in account that
and put
then the penultimate of the equalities (1) gives the following:
Substituting here the expression of
by
and
from the preceding of the equalities (1), we obtain:
where obviously
,
.
Continuing rising up on the equalities (1) we come at last to the
required equality (2).
To prove the second assertion of the theorem
suppose that
and
satisfying the equality (2) have been already found, but for example
the degree of
is greater than or equal to the degree of
.
Divide
on
:
where the degree of
is less than the degree of
,
and substitute this expression in (2). We obtain:
The degree of
is less than the degree of
.
The degree of the polynomial standing in square brackets will be less
than the degree of
since otherwise the degree of the second summand of the left part
would be no less than the degree of product
.
Then all the left part would have the degree that is greater than or
equal to the degree of
but
has at our assumptions smaller degree.
Corollary 2.
Polynomials
and
are relatively prime iff it can be found polynomials
and
satisfying the equality
.