# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 6

*Polynomials*

Mathematical mappings are usually given by a mathematical term, an expression which describes how to get (compute) from a given number its value. Here we consider polynomial functions, which are built in an easy manner. Their definition and basic properties work over any field.

The numbers are called the *coefficients* of the polynomial. Two polynomials are equal if all their coefficients coincide. The polynomials with
for all
are called *constant polynomials*, we write simply for them. In the *zero polynomial*, all coefficients equal . Using the sum symbol, one can write a polynomial briefly as .

## Definition

The *degree* of a nonzero polynomial

with

is .The zero polynomial does not have a degree. The coefficient , where is the degree of the polynomial, is called the *leading coefficient* of the polynomial. The term is called the *leading term* of the polynomial.

The set of all polynomials over a field is called *polynomial ring* over , it is denoted by , where is the *variable* of the polynomial ring.

Two polynomials

can be added by adding the components, i.e. the coefficients of the sum are just the sums of the coefficients of the two polynomials. In case , the "missing“ coefficients of are to be interpreted as . This addition is obviously associative and commutative, the zero polynomial is the neutral element and the negative polynomial is obtained by taking from every coefficient of its negative.

One can also multiply two polynomials, one puts

and extends this multiplication rule so that distributivity holds. This means one multiplies each summand with each summand and adds then everything together. One can describe this multiplication also explicitly by the rule:

For the degree the following rules hold.

If a polynomial
and an element
are given, then one can *insert* into meaning that one replaces everywhere the variable by . This yields a mapping

which is called the corresponding *polynomial function*.

If and are polynomials, then the composition is described in the following way: one has to replace everywhere in the variable by and then simplify this expression. The result is again a polynomial. The order of and is important for this.

*Euclidean division*

If we have a polynomial over the reals, we are interested in its zeros, its growth behavior, local maxima and minima. For these questions, the euclidean division for polynomials (long division) is important.

## Theorem

Let be a field and let be the polynomial ring over . Let be polynomials with . Then there exist unique polynomials such that

### Proof

We prove the statement about the existence by induction over the degree of . If the degree of is larger than the degree of , then and is a solution.

Suppose that . By the remark just made also holds, so is a constant polynomial, and therefore (since and is a field) and is a solution.

So suppose now that and that the statement for smaller degrees is already proven. We write and with . Then setting we have the relation

The degree of this polynomial is smaller than and we can apply the induction hypothesis to it. That means there exist and such that

From this we get altogether

so that and is a solution.

To prove uniqueness, let , both fulfilling the stated conditions. Then . Since the degree of the difference is smaller than , this implies and so .

The computation of the polynomials
and
is also called *long division*. The polynomial is a factor of if and only if the division of by yields the remainder . The proof of this theorem is constructive, meaning it can be used to do the computation effectively. For this, one has to be able to do the computing operations in the field . We give an example.

## Example

We want to apply the Euclidean division (over )

So we want to divide a polynomial of degree by a polynomial of degree , hence the quotient and also the remainder have (at most) degree . For the first step, we ask with which term we have to multiply to achieve that the product and have the same leading term. This is . The product is

The difference between and this product is

We continue the division by with this polynomial, which we call . In order to get coincidence with the leading coefficient we have to multiply with . This yields

The difference between this and is therefore

This is the remainder and altogether we get

## Lemma

Let be a field and let be the polynomial ring over . Let be a polynomial and . Then is a zero of if and only if is a multiple of the linear polynomial .

### Proof

If is a multiple of , then we can write

with another polynomial . Inserting yields

In general, there exists, due to Theorem 6.3 , a representation

where either or the degree of is , so in both cases is a constant. Inserting yields

So if holds, then the remainder must be , and this means .

## Corollary

Let be a field and let be the polynomial ring over . Let be a polynomial () of degree . Then has at most zeroes.

### Proof

We prove the statement by induction over . For the statement holds. So suppose that and that the statement is already proven for smaller degrees. Let be a zero of (if does not have a zero at all, we are done anyway). Hence, by Lemma 6.5 and the degree of is , so we can apply to the induction hypothesis. The polynomial has at most zeroes. For we have . This can be zero, due to Lemma 5.5 , only if one factor is , so the zeroes of are or a zero of . Hence, there are at most zeroes of .

*Fundamental theorem of algebra*

The following *fundamental theorem of algebra*, for which we do not provide a proof, shows the importance of complex numbers.

## Theorem

Every nonconstant polynomial over the complex numbers has a zero.

The fundamental theorem of algebra implies that for every polynomial
,
there is a factorization into linear factors, i.e. one can write

with uniquely determined complex numbers and (where repetitions are allowed).

*Polynomial interpolation*

The following theorem is called *theorem about polynomial interpolation* and describes the interpolation of given function values by a polynomial. If just one function value at one point is given, then this determines a constant polynomial, two values at two points determine a linear polynomial
(the graph is a line),
three values at three points determine a quadratic polynomial, etc.

## Theorem

Let be a field, and let different elements and elements are given. Then there exist a unique polynomial of degree , such that holds for all .

### Proof

We prove the existence and consider first the situation where for all for some fixed . Then

is a polynomial of degree which at the points has value . The polynomial

has at these points still a zero, but additionally at its value is . We denote this polynomial by . Then

is the polynomial looked for, because for the point we have

for and .

The uniqueness follows from Corollary 6.6 .

## Remark

If the data and are given, then one can find the interpolating polynomial of degree , which exists by Theorem 6.8 , in the following way: We write

with unknown coefficients , and determine then these coefficients. Each interpolation point yields a linear equation

over . The resulting system of linear equations has exactly one solution , which gives the polynomial.

We will deal with systems of linear equations later in more detail, it should however be known from school how to find the solutions.

*Rational functions*

Next to the polynomial functions, the simplest functions are the rational functions.

## Definition

For polynomials , , the function

where is the complement of the zeroes

of , is called a*rational function*.

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