Master the Basics of Polynomials

Polynomials are expressions consisting of indeterminates and coefficients that involve only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. Polynomials appear in many areas of mathematics and science, including polynomial equations, polynomial functions, calculus, and numerical analysis. The word polynomial joins the Greek poly and Latin nomen, meaning "many names," as it is a sum of many terms. The x occurring in a polynomial is commonly called a variable or an indeterminate. Polynomials can be denoted either as P or P(x), and the functional notation P(x) is often useful for specifying a polynomial and its indeterminate. A polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. The degree of a polynomial is the largest degree of any term with a nonzero coefficient. Polynomials of small degree have specific names, such as linear, quadratic, and cubic polynomials. The zero polynomial is unique in that it is the only polynomial in one indeterminate that has an infinite number of roots, and its degree is either left undefined or defined as negative. Polynomials can be classified by the number of terms with nonzero coefficients, such as monomial, binomial, and trinomial. A real polynomial is a polynomial with real coefficients, and an integer polynomial is a polynomial with integer coefficients. A polynomial in one indeterminate is called a univariate polynomial, while a polynomial in more than one indeterminate is called a multivariate polynomial.Overview of Polynomials

• Polynomials are mathematical expressions consisting of variables and coefficients, combined using arithmetic operations such as addition, subtraction, multiplication, and division.

• A polynomial with one variable is called a univariate polynomial; with two variables, it is called a bivariate polynomial. Polynomials can be classified further as multivariate polynomials based on the maximum number of indeterminates allowed.

• The evaluation of a polynomial involves substituting numerical values for each variable and carrying out the arithmetic operations.

• Polynomials can be added and subtracted using the associative and commutative laws of addition, and combined like terms. Multiplication of polynomials involves the distributive law applied repeatedly.

• The composition of two polynomials involves substituting each copy of the first variable with the second polynomial. Division of polynomials is not typically a polynomial, but rather a rational fraction, expression, or function.

• All polynomials with coefficients in a unique factorization domain have a unique factored form, which is a product of irreducible polynomials and a constant. Polynomial functions are functions defined by evaluating a polynomial.

• A polynomial function is continuous, smooth, and entire. The graph of a non-constant polynomial function tends to infinity when the variable increases indefinitely.

• A polynomial equation is an equation of the form ax^n + bx^(n-1) + ... + c = 0, where the unknowns are the variables of the polynomial. The solutions are the possible values of the unknowns for which the equality is true.Polynomials: A Summary

• The number of solutions of a polynomial equation with real coefficients equals the degree when the complex solutions are counted with their multiplicity, and this fact is known as the fundamental theorem of algebra.

• A root of a nonzero univariate polynomial P is a value of x such that P(a) = 0, and a power greater than 1 of x - a divides P if a is a multiple root of P.

• The coefficients of a polynomial and its roots are related by Vieta's formulas.

• Every non-constant polynomial has at least one root when the set of accepted solutions is expanded to the complex numbers, and the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

• There are formulas for solving equations of degrees one and two, and since the 16th century, similar formulas are known for equations of degree three and four. However, there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula involving only arithmetic operations and radicals.

• For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called zeros instead of "roots", and the study of the sets of zeros of polynomials is the object of algebraic geometry.

• A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation, and solving Diophantine equations is generally a challenging task.

• Trigonometric polynomials are finite linear combinations of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers.

• A matrix polynomial is a polynomial with square matrices as variables, and a matrix polynomial equation is an equality between two matrix polynomials which holds for the specific matrices in question.

• A rational fraction is the quotient of two polynomials, and any algebraic expression that can be rewritten as a rational fraction is a rational function.

• Laurent polynomials are like polynomials but allow negative powers of the variable(s) to occur.

• Formal power series are like polynomials but allow infinitely many non-zero terms to occur, and the rules for manipulating their terms are the same as for polynomials.Overview of Polynomials

• Polynomials and polynomial functions are often used interchangeably in analysis.

• Distinguishing between the two is important for operations such as Euclidean division.

• If R is an integral domain, f divides g if there exists a polynomial q in R[x] such that f q = g.

• If F is a field and g ≠ 0, then there exist unique polynomials q and r in F[x] with f = gq + r.

• Prime polynomials are non-zero polynomials that cannot be factored into the product of two non-constant polynomials.

• Polynomials are used in modern positional number systems as shorthand notation.

• Polynomial functions are useful for polynomial approximations, such as Taylor's theorem and the Stone-Weierstrass theorem.

• Polynomials are used to encode information about other objects, such as matrices, linear operators, and graphs.

• The term "polynomial" is also used for quantities or functions that can be written in polynomial form.

• The roots of polynomials have been studied for thousands of years, but modern notation only developed in the 15th century.

• René Descartes introduced the concept of the graph of a polynomial equation and popularized the use of letters to denote constants and variables, as well as superscripts for exponents.