Humâyûn

In Algebra, a function Polynomial , or polynomial function is defined as being a application associated with a Polynôme to coefficients in a ring (often a body) commutative $K$ of the form:

$f : X \ mapsto a_n x^n + a_ \left\{N - 1\right\} x^ \left\{N - 1\right\} + \ cdots + a_1 X + a_0 x^0$

where $n$ is a natural entirety and $a_n$, $a_ \left\{N - 1\right\}$,…, $a_0$ are elements of $K$, called Coefficient S of the function polynomial $f$. That is still written, using the notation sigma:

$f: X \ mapsto \ sum_ \left\{R = 0\right\} ^ \left\{N\right\} a_r x^ \left\{R\right\}$
It is said that $f$ is a function polynomial with coefficients in $K$.

One did not specify the starting $K$ whole and arrival $L$ of a function polynomial in order not to complicate the definition. It is enough in fact that $L$ is provided with a Structure of algebra on the body (or the ring) $K$. Such a structure comprises all the operations which intervene in the definition of a function polynomial:

• the internal laws of multiplication and addition of the ring K make it possible to multiply and add the coefficients between them.

• a external law of multiplication makes it possible to make the product of an element of the ring K and of an element of a unit L.
• an internal law of multiplication makes it possible to make the product of element X with itself as a whole L.
• an internal law of addition makes it possible to add between them the elements of the form $a_k x^k$ pertaining to L.

In practice, one often places oneself in the particular cases $K=L= \ mathbb R$ (or $K=L= \ mathbb C$) in which all the laws of preceding multiplications are confused.

In analyzes, one almost always considers functions polynomials with coefficients real or complex ($K= \ mathbb R$ or $K= \ mathbb C$).

Degree

The degree of a polynomial function $f$ nonnull is largest of the natural entireties $k$ such as $a_k$ is nonnull (it is thus $n$ if the coefficient $a_n$ is not null). By convention, the degree of the null polynomial function is $- \ infty$.

Each term of the function polynomial of the form $a_k x^k$ is called a students' rag procession (of $k$ degree). The coefficient of the students' rag procession moreover high degree is called the coefficient dominating of $f$; $a_0$ is called the constant coefficient of $f$.

Identification of the coefficients

If $K$ is an infinite commutative body, there is equivalence between the formal identity polynomials with coefficients in $K$ and the identity of the associated functions polynomials: two polynomials are equal (even degree and same coefficients have) if and only if the associated polynomial functions are equal.

In more abstract terms: the Morphisme of K-algebras $P \ mapsto \ tilde \left\{P\right\}$ of K in $\ mathcal F \left(K\right)$ which with a polynomial $P \left(X\right) = \ sum_ \left\{R = 0\right\} ^ \left\{N\right\} a_r X^r$ of K associates the polynomial function $\ tilde \left\{P\right\}: K \ to K, \, X \ mapsto \ sum_ \left\{R = 0\right\} ^ \left\{N\right\} a_r x^r$, is then injective.

In this case, it is necessary no more to distinguish the polynomial and the associated polynomial function.

Particular polynomials

The polynomials of

• degree 0 are called constant functions nonnull,
• degree 1 (or $\ leq$ 1) are called functions closely connected ,
• degree 2 are called quadratic functions ,
• degree 3 are called cubic functions

The function polynomial $f \left(X\right) = -7x^3 + \ frac \left\{2\right\} \left\{3\right\} x^2 - 5x + 3$ is an example of a cubic function with like coefficient dominating -7 and constant coefficient 3.

Importance of the functions polynomials

The functions polynomials are often used because they are the simplest functions: their definition implies only the addition and the multiplication (since the powers are only shorthand for the repeated multiplications).

They are also simple in another direction: the polynomials of degree lower or equal to N are precisely the functions whose Dérivée ( N +1) ième is identically null.

An important aspect in numerical calculation is the possibility of studying the complicated functions by means of approximations by polynomials. Theorems make possible such studies under certain conditions.

Most important are the Théorème of Taylor, which affirms about that any function differentiable N time with the air to be locally a polynomial, and the theorem of approximation of Weierstrass, which affirms that any function continues definite on a compact interval can be uniformly approximate on this interval of also close wished by a polynomial.

The Quotient S of functions polynomials are called the rational functions. Those are the only functions which can be evaluated directly by a Ordinateur, since at the base, only the operations of addition, multiplication and division (and logical operations) can be carried out by the central processing unit of a Ordinateur. All the other functions which one has need to evaluate using a computer, like the goniometrical functions, the exponential functions logarithms and functions, must be then approximate by suitable rational functions.

To evaluate functions polynomials in values given of the variable X , one does not apply the polynomial like a formula and one does not calculate all the powers of X , but one uses rather the Méthode of Horner, much more effective.

If the evaluation of a polynomial in many equidistant points is required, then the method the finite differences of Newton reduces the quantity of spectacular work of way. The engine of differences of Charles Babbage was designed automatically to create large tables of values of the functions logarithms and trigonometrical by evaluating polynomials with the method of the differences in Newton, by using many points.

Roots

A root or a zero of a polynomial P ( X ) is a number R such as P ( R ) = 0. To determine the roots of a polynomial of degree equal to or higher than 1, or “to solve an algebraic equation”, is one of oldest mathematical problems the. Certain polynomials, like $P \left(X\right) = X^2 + 1$, do not have a root as a whole of the real numbers. If the roots are required in the whole of the complex numbers, then one will be able to find at least one of them (here, there are two of them.) Indeed any polynomial (not-constant) of $\ mathbb C$ admits at least a root complexes (see the Théorème of Alembert-Gauss.)

Order of multiplicity of a root

If R is root of the polynomial P (X) , there exists a polynomial Q (X) such as $P \left(X\right) = \left(X-r\right) Q \left(X\right)$ (to show it is enough to cut off with each students' rag procession $a_k X^k$ to $P \left(X\right)$ the value $a_k r^k$ and to note that $\left(X-r\right)$ is put naturally in factor). If $Q \left(R\right)$ is null, then one can still put $\left(X - R\right)$ in factor. It is said whereas R is root double of P (X).

More generally, if there exists a polynomial Q (X) and a natural entirety not no one m such as $P \left(X\right) = \left(X-r\right) ^m Q \left(X\right)$ and Q (R) ≠0 , it is said that R is root of order m , or has as a multiplicity m ( Q and m is then single). For example, the polynomial $P \left(X\right) = X^3 - 2X^2 + X$ can be also written $P \left(X\right) = \left(X-1\right) ^2 X$; thus 1 is a root of P , and its multiplicity is equal to 2, whereas 0 are simple root .

Calculation of the roots of a polynomial

The research of the roots of the polynomials of degree 1 or 2 are traditional teaching pre-academic, known like " solution of an equation of the first or the second degré". Formulas making it possible to calculate the roots of the polynomials of degree up to 4 starting from the coefficients by using the four arithmetic operations plus the radicals (nth roots) were already known at the sixteenth century (formula of Cardan joint, of Nicolas Fontana Tartaglia, Ferrari).

No general formula of this type exists for the polynomials of degree 5 or more, like Abel in 1824 proved it. This result precedes by little the more general theory developed by Welsh which engages in a detailed study of the relations between the roots of polynomials.

The approximations of the real roots of a given polynomial can be found by using the Méthode of Newton, or more effectively by using the Méthode of Laguerre which employs the arithmetic complex and makes it possible to locate all the complex roots. These algorithms are studied in numerical Analyze

See too

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