Zero of a function

The zero is a term from the field of mathematics which deals with functions and their gradients and properties. Is understood as zeros of those values ​​that provide the function value of zero used in a function. The word ' body' indicates this is that there are elements of the domain. For real functions are exactly the points of the axis on which the graph of a function touches or intersects the axis. Zeros of polynomials are referred to as roots.

• 2.1 Determination of the zeros of polynomials of small degree
• 2.2 polynomials with integer coefficients
• 2.3 polynomials with real coefficients
• 2.4 polynomials with only real zeros
• 2.5 polynomials with complex coefficients
• 2.6 Polynomials over full rated bodies

Zeros of real valued functions

Definition

One element of defining amount of a function is called a zero of if applies. One then also say has a zero at or disappears at the point

Example

The values ​​3 and -3 are zeros of the function, because and.

The value 0 is not zero, because.

Multiple zeros

Is a polynomial or at least continuous and differentiable at the zero point, so you can " share out " the zero point. More precisely: There is a continuous function in, such that for all.

There are then two cases:

To determine whether a single or multiple zero is to use the fact that the value is equal to the derivative of the position. So, for a differentiable function, one gets the following criterion:

If is often differentiable, then you can repeat this process. One defines:

It is a natural number. A - times differentiable function on an open subset has a ( at least ) times zero or a zero of order (at least) if itself and the first derivatives of at the point take the value zero:

Is one times zero, but no times, so

It is called the order or multiplicity of the zero point.

Example

With the derivatives

It is true, that is a zero of. Next is valid

Thus, 1 is a triple but no four-fold zero of, so a zero of multiplicity 3

Other properties

• A function has exactly then at one times zero, if a zero and one times zero is at.
• A - times continuously differentiable function has exactly then at least one zero at times when there is a continuous function such that

• A - times continuously differentiable function has exactly then at a zero of multiplicity if there exists a continuous function such that

• The function

At 0 has a zero of order infinity, see also Analytical function.

Existence and computation of zeros

From the intermediate value theorem, one can often indirectly infer the existence of a zero point: If two function values ​​of a continuous function of a positive and a negative, has at least one zero between and. ( Illustratively stated, the functional graph connecting the two points and which intersect the axis. )

Depending on the function, it may be difficult or impossible to determine explicitly the zeros, ie the equation

According to dissolve. In this case one can approximate values ​​for nulls using various numerical methods, such as the bisection ( bisection method ), Regula falsi or an appropriate fixed-point iteration for continuous functions, Newton's method or Halley method for differentiable functions, the Weierstrass (Durand - Kerner ) determine process or the Bairstow method for polynomials.

In the list of numerical methods to find the root-finding under the section Nonlinear systems of equations.

Roots of polynomials

Is a ring and a polynomial over, it means an element zero of if the setting of zero results in:

Is a ring homomorphism, so analog zeros can be defined from to.

Using the polynomial, one can show that if and a zero of is when is divisible by, so if there is a polynomial such that

Applies. This statement is sometimes called a Nullstellensatz; However, there is a likelihood of confusion with the hilbert between Nullstellensatz.

A times zero or zero of order is an element, so that is divisible by. It is also called the multiplicity or multiplicity of the zero.

Determination of the zeros of polynomials small degree

For polynomials over a field of degree at most four, there is general solution formulas with radicals to determine the zeros directly:

• Grade 1: see linear equation. The polynomial has the zero point.
• Grade 2: see quadratic equation
• Grade 3: see cubic equation
• Grade 4: see quartic equation

Polynomials with integer coefficients

Is a polynomial with integer coefficients, each integer is a divisor of zero.

From the lemma of Gauss follows: If a monic polynomial with integer coefficients, then every rational zero is an integer and hence a divisor of.

Example:

The divisor of the absolute element of are no zeros, so it has no rational zeros. Since each factorization would contain of a linear factor, it follows that is irreducible.

Polynomials with real coefficients

Polynomials of odd degree over the real numbers always have at least one real zero; this follows from the intermediate value theorem. Another argument (if one already has available the fundamental theorem of algebra) is the following: real complex roots of real polynomials always occur as pairs of conjugate complex numbers on. So even or odd degree polynomials have always even or odd lots of real zeros, if you count each zero according to their multiplicity. An application of the latter principle represents the numerical Bairstow method

Example:

The polynomial has the zero point, which can be easily guessed as a divisor of the absolute element. This is obtained by polynomial

Resulting in even the two mutually complex conjugated zeros and devoted.

Polynomials with only real zeros

Is a polynomial whose zeros are all real, then these are in the interval with endpoints

Example:

The polynomial has four real zeros -3, -2, -1 and 1 using the interval formula gives

Rounded, the interval

The zeros are therefore located in the interval of found.

Polynomials with complex coefficients

The fundamental theorem of algebra states that every non-constant polynomial over the complex numbers has at least one zero. By repeating cleaves linear factors to zeros, we obtain the statement that every polynomial

On the complex numbers of the form

Write leaves. Here are the different zeros of and their respective multiplicities.

Polynomials over full rated bodies

It is a fully weighted body with valuation ring and residue field, and it is a monic polynomial. It follows from the hensel between Lemma: If the reduction of a simple zero in such a zero has.

Example:

It is the body of the p- adic integers for a prime number. Then and. The polynomial is divided into different linear factors, so it also has about exactly zeros, ie contain -th roots of unity.

• Mathematical concept