What Is a Polynomial?

In mathematics, an algebraic expression consisting of the addition of several mononomials is called a polynomial (if there is a subtraction: subtracting a number is equal to adding its opposite number). Each mononomial in a polynomial is called a polynomial term, and the highest degree of these mononomials is the degree of this polynomial. The term without a letter in the polynomial is called a constant term.

In mathematics, polynomial means

Polynomial is simple

Polynomial functions and their roots

Give a polynomial fR [x ₁ , ..., x _n ] and an R-algebra A. For (a ₁ , ..., an _n ) An, we replace all x _j in f with a _j to get an element in A, and write it as f (a ₁ ... an _n ). In this way, f can be seen as a function from An to A.

If f (a ₁ ... a _n ) = 0, then (a ₁ ... a _n ) is called the root or zero of f.

For example, f = x ^ 2 + 1. If we consider that x is a real number, a complex number, or a matrix, then f will have no roots, two roots, and infinite roots!

For example, f = xy. If we consider that x is real or complex, then the set of zeros for f is the set of all (x, x) and is an algebraic curve. In fact all algebraic curves come from this.

In addition, if all coefficients are real-number polynomials P (x) have complex roots Z, the common-orbit complex number of Z is also the root.

If P (x) has n overlapping roots, then P '(x) has n-1 overlapping roots. That is, if P (x) = (xa) ^ nQ (x), then a is the overlapping root of P '(x) and there are n-1. ^[2]

Polynomial interpolation polynomial

In practical problems, the quantitative relationship y = F (x) that indicates a certain rule is often obtained through experiments or observations. Usually, only the function value of F (x) at some points xi is given as yi = F (xi). , J = 1, 2, ..., n + 1. Even if the analytic expression of the function F (x) is sometimes given, it is not easy to calculate if it is more complicated. Therefore, according to the function value F (xi) at a given point xi, a simple function (x) that can reflect the characteristics of F (x) and is easy to calculate is needed to approximate F (x). At this time, (x) is called F (x) 's interpolation function; x1, x2, ..., xn + 1 are called interpolation nodes. The method of finding the interpolation function is called interpolation.

Polynomials are a class of simple elementary functions, and they can be assigned to two sets of numbers: b1, b2,, bn + 1 and different 1, 2,, n + 1. There are always unique polynomials that do not exceed n (x) satisfies (i) = bi, i = 1, 2, ..., n + 1. Therefore, in practical applications, polynomials are often used as interpolation functions. The polynomial as an interpolation function is called an interpolation polynomial. Interpolation polynomials are most commonly used in computing mathematical interpolation. ^[2]