3.4 Komplexe Polynome

Aus Online Mathematik Brückenkurs 2

Version vom 14:08, 5. Sep. 2008

Polynom and equations

An expression of the form

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where \displaystyle n is an integer, is called a polynomial of degree \displaystyle n in an unknown variable \displaystyle x. The number \displaystyle a_1 is called the coefficient of \displaystyle x, \displaystyle a_2 the coefficient of \displaystyle x^2, etc. The constant \displaystyle a_0 is called the constant term.

Polynomials are essential for a large part of mathematics and have many properties which display great similarities with our integers, which means that we can do calculations with polynomials in a similar way as with integers.

If \displaystyle p(x) is a polynomial of degree \displaystyle n then \displaystyle p(x)=0 is called a polynomial equation of degree \displaystyle n. If \displaystyle x=a is a number such that \displaystyle p(a)=0 then \displaystyle x=a is called a root, or solution to the equation. One also says that \displaystyle x=a is a zero of \displaystyle p(x).

As the above example showed polynomials can be divided just like integers. Polynomial division, just like integer division is usually not exact. If, for example, \displaystyle 37 is divided by \displaystyle 5, one gets

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The calculation can also be written as \displaystyle \ 37= 7\cdot 5+2\,. The number 7 is called the quotient and the number 2 the remainder. One says that dividing 37 by 5 gives the quotient 7 and the remainder 2.

If \displaystyle p(x) and \displaystyle q(x) are polynomials one similarly can divide \displaystyle p(x) with \displaystyle q(x) and unambiguously determine polynomials \displaystyle k(x) and \displaystyle r(x) such that

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or \displaystyle \ p(x)= k(x)\cdot q(x)+r(x)\,. One here says that polynomial division has resulted in a quotient \displaystyle k(x) and remainder \displaystyle r(x).

It is obvious that a division is exact if the remainder is zero. For polynomials this is expressed as follows: If \displaystyle r(x)=0 then \displaystyle p(x) is divisible by \displaystyle q(x), or, \displaystyle q(x) is a divisor of \displaystyle p(x). One writes

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or \displaystyle \ p(x) = k(x)\cdot q(x)\,.

Polynomial division

If \displaystyle p(x) is a polynomial of higher degree than polynomial \displaystyle q(x) then one can divide \displaystyle p(x) by \displaystyle q(x). For example, this may be done by gradually subtracting appropriate multiples of \displaystyle q(x) from \displaystyle p(x) until a remaining numerator is of lower degree than the denominator \displaystyle q(x).

The connection between factors and zeros

If \displaystyle q(x) is a divisor of \displaystyle p(x) then \displaystyle p(x)=k(x)\cdot q(x). We have thus factorised \displaystyle p(x) . One says that \displaystyle q(x) is a factor of \displaystyle p(x). Especially, if a polynomial of first degree \displaystyle (x-a) is a divisor of \displaystyle p(x) then \displaystyle (x-a) is a factor of \displaystyle p(x) , i.e.

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Since \displaystyle \ p(a)=q(a)\cdot (a-a)= q(a)\cdot 0 = 0\ this means that \displaystyle x=a is a zero of \displaystyle p(x). This is exactly the content of the so-called factor theorem.

Please note that the theorem applies in both directions, i.e. if we know that \displaystyle x=a is a zero of \displaystyle p(x) we automatically would know that \displaystyle p(x) is divisible by \displaystyle (x-a).

Fundamental theorem of algebra

We introduced at the beginning of this chapter, the complex numbers to enable us to solve the quadratic equation \displaystyle x^2=-1 and we can now ask ourselves the slightly more theoretical question, whether this is sufficient, or do we need to invent more types of numbers in order to solve other complicated polynomials? The answer to that question is that we need not as the complex numbers are enough. The German mathematician Carl Friedrich Gauss proved in the year 1799 the fundamental theorem of algebra which says the following:

As every zero according to the the factor theorem is matched by a factor, we can now also state the following theorem:

(By multiplicity is meant that a double zero is counted twice, a triple zero 3 times, etc.)

Note that these theorems only say that there exists complex roots of a polynomial, but not how to determine them. In general, there is no simple method to write a formula for the roots, and for polynomials of higher degree, we must use various devices to obtain a solution. If we restrict ourselves to polynomial with real coefficients, one of the devices that can help us is the knowledge that the complex roots of such polynomials always come in complex conjugate pairs.

One consequence of the fundamental theorem of algebra (and the factor theorem) is that all polynomials can be factored into a product of complex first order factors. This also applies to polynomials with real coefficients, but for such polynomials it is possible to multiply together the pair of factors belonging to complex conjugate roots. In this case the factorisation will consist of first and second order real factors.