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 coefficientis 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 andareequal (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 andwith complex coefficients written for convenience by increasing degrees ofare given:

and if for example , theirsum is the polynomial

of which the coefficients are obtained by adding the coefficients of polynomials andat the same degrees of, i.e..

Let non-zero polynomials andwith complex coefficients be given. If the remainder of dividingonis equal to zero, i.e.is divided onwithout any remainder, then the polynomialis called adivisor of the polynomial . For example, the polynomialis a divisor of the polynomial, andis a divisor of.

A polynomial is acommon divisor for andif it is a divisor for each of these polynomials. Obviously, all the polynomials of zero degree are common divisors ofand. If there are no other divisors for them, they are calledrelatively prime. For example, the polynomials andare relatively prime.

The greatest common divisor (gcd) of non-zero polynomials andis called such a polynomialwhich is their common divisor and it is divided on any other common divisor of these polynomials. The greatest common divisor ofandis denoted by. For example, the polynomialis the greatest common divisor of the polynomialsand.

29. Polynomials: the Euclid algorithm (the algorithm of sequential dividing).

Let andbe polynomials. Divideonand we obtain some remainderin general. Then divideonand we obtain a remainder, divideonand 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 remainderthat divides the preceding remainderwill be the greatest common divisor of the polynomialsand.

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 thereforewill be a divisor for. Further rising up by the same way we obtain thatis a divisor for, …,,. Then by the second equalityis a divisor for, and therefore by the first equalityis a divisor for. Thus,is a common divisor forand.

Take now an arbitrary common divisor of the polynomialsand. 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 thatis a divisor of,, … At last if it will be proved thatandare divide onthen from the penultimate equality we obtain thatis divided on. Thus,will be gcd forand.

If is gcd forandthen we can choose as gcd the polynomialwhereis 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 ofandthen it can be found such polynomialsandthat(2)

We can assume that if the degrees of andis greater than zero then the degree ofis less than the degree of, and the degree ofis less than the degree of.

Proof: The proof is based on the equalities (1). If we take in account that and putthen the penultimate of the equalities (1) gives the following:

Substituting here the expression of byandfrom 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 andsatisfying the equality (2) have been already found, but for example the degree ofis greater than or equal to the degree of. Divideon:where the degree ofis 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 ofsince 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 ofbuthas at our assumptions smaller degree.

Corollary 2. Polynomials andare relatively prime iff it can be found polynomialsandsatisfying the equality.