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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
30. Roots of polynomials. Horner’s method. Basic theorem and its consequences
If
is some polynomial, and
is a number then the number
obtained by replacing the unknown
on
is called thevalue of the polynomial
at
.
If
,
the number
is called aroot of
the polynomial
(or the equation
).
If we divide a polynomial
on an arbitrary polynomial of the first degree (or as we say further
alinear polynomial)
then the remainder will be either some polynomial of the zero degree
or zero, i.e. in any case some number
.
Theorem 3. The
remainder of dividing a polynomial
on a linear polynomial
is equal to the value
of the polynomial
at
.
Proof: In fact,
let
.
Taking the values of both parts of this equality at
,
we obtain:
.
Corollary 4. A
number
is a root of a polynomial
iff
divides
.
On other hand, if
is divided on some polynomial of the first degree
then obviously it also is divided on the polynomial
,
i.e. on a polynomial of the form
.
Thus, finding roots of a polynomial
is equivalent to finding its linear divisors.
Horner’s method
Let
,
and let
(*)
where
.
Comparing the coefficients at the same degrees of
in (*), we obtain:
……………
Then
.
At last
.
Thus, the coefficients of the quotient and the remainder can be
sequentially obtained by calculations of the same type.
Basic theorem and its consequences
Theorem 5 (Basic theorem of algebra of complex numbers) Every polynomial with arbitrary numeric coefficients of which the degree is greater than or equals 1 has at least one root (in general, it is complex).
Let
be a polynomial of the
th
degree,
with arbitrary complex coefficients. The basic theorem allows to
assert an existence of a root
for
.
Therefore,
.
The coefficients of
are also real or complex numbers, and therefore
has a root
,
and consequently
.
Continuing by such way we come after a finite
number of steps to a decomposition of the polynomial of the
th
degree into product of
linear multipliers:
(1)
The decomposition (1) is unique for a polynomial
up to the order of multipliers.
Indeed, if there is another decomposition:
(2)
then from (1) and (2) follows the following:
(3)
If the root
would be different from all
then substituting
instead of the unknown in (3) we would be obtained on the left zero
and on the right a number differed from zero. Thus, each root
is equal to some root
and conversely. This doesn’t imply coinciding (1) and (2). Indeed,
among the roots
can be equal each other. Let for example there are
roots equal to
and on the other hand there are
roots among
that are equal to
.
It needs to show that
.
If for example
then reducing both parts of (3) on the multiplier
we come to the equality of which the left part contains the
multiplier
,
and the right part doesn’t contain it. It is a contradiction. Thus,
the decomposition (1) for
is unique.
Joining together identical multipliers, the decomposition (1) can be rewritten as follows:
(4)
where
,
and all
are distinct.
We proved the following important result:
Theorem 6. Every
polynomial
of the degree
,
with arbitrary numeric coefficients has
roots if each of the roots is counted as many as its multiplicity.