A polynomial, or polynomial, is one of the basic algebraic structures found in school and higher mathematics. The study of the polynomial is the most important topic in the course of algebra, since on the one hand polynomials are quite simple in comparison with other types of functions, and on the other they are widely used in solving problems of mathematical analysis. So what is a polynomial?
Definition
The term polynomial can be defined through the concept of a monomial, or monomial.
A monomial is an expression of the form cx 1 i1 x 2 i2 ... x n in . Here c is a constant, x 1 , x 2 , ... x n are variables, i1, i2, ... in are exponents of the variables. Then the polynomial is any finite sum of monomials.
To understand what a polynomial is, you can look at specific examples.
The quadratic trinomial, examined in detail in the 8th grade math course, is a polynomial: ax 2 + bx + c.
A polynomial with two variables may look like this: x 2 -x + y 2 . Such a polynomial is also called an incomplete square of the difference x and y.
Polynomial Classifications
By degree of polynomial
For each monomial, the sum of exponents i1 + i2 + ... + in is found in the composition of the polynomial. The largest of the sums is called the polynomial exponent, and the monomial corresponding to this sum is called the senior term.
By the way, any constant can be considered a polynomial of degree zero.
Reduced and unreduced polynomials
If the leading term has coefficient c equal to 1, then the polynomial is reduced; otherwise, it is not.
For example, the expression x 2 + 2x + 1 is the reduced polynomial, and 2x 2 + 2x + 1 is the non-reduced.
Homogeneous and heterogeneous polynomials
If the degrees of all members of the polynomial are equal, then they say that such a polynomial is homogeneous. All other polynomials are considered heterogeneous.
Homogeneous polynomials: x 2 -hu + y 2 , xyz + x 3 + y 3 . Inhomogeneous: x + 1, x 2 + y.
There are special names for a polynomial of two and three members: binomial and trinomial, respectively.
Polynomials of one variable are distinguished into a separate category.
Using a polynomial in one variable
Polynomials of one variable approximate continuously continuous functions of varying complexity from one argument.
The fact is that such polynomials can be considered as partial sums of a power series, and a continuous function can be represented as a series with an arbitrarily small error. Function expansion series are called Taylor series, and their partial sums in the form of polynomials are called Taylor polynomials.
Studying graphically the behavior of a function by approximating it with some polynomial is often easier than exploring the same function directly or using a series.
It is easy to search for derivatives of polynomials. For finding the roots of polynomials of degree 4 and below, ready-made formulas exist, and approximate algorithms of high accuracy are used to work with higher degrees.
There is also a generalization of the described polynomials for functions of many variables.
Binomial theorem
Famous polynomials are Newton polynomials, deduced by a scientist to find the coefficients of the expression (x + y) n .
It is enough to look at the first few degrees of the binomial expansion to verify that the formula is non-trivial:
(x + y) 2 = x 2 + 2hu + y 2 ;
(x + y) 3 = x 3 + 3x 2 y + 3hu 2 + y 3 ;
(x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4x 3 + y 4 ;
(x + y) 5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5hu 4 + y 5 .
For each coefficient, there is an expression that allows it to be calculated. However, memorizing cumbersome formulas and performing the necessary arithmetic operations each time would be extremely inconvenient for those mathematicians who often require such decompositions. Pascal’s triangle made life much easier for them.
The figure is built according to the following principle. At the top of the triangle is written 1, and in each next line it becomes one more digit, 1 is placed at the edges, and the middle of the line is filled with the sums of two adjacent numbers from the previous one.
When you look at the illustration, everything becomes clear.
Of course, the above examples, the most widely known, the application of polynomials in mathematics is not limited.