The constant polynomial whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials.
The degree of the zero polynomial is undefined, but many authors conventionally set it equal to or . In Mathematica, Exponent[0, x] returns -Infinity.
Variable
A variable is a symbol on whose value a function, polynomial, etc., depends. For example, the variables in the function are and . A function having a single variable is said to be univariate, one having two variables is said to be bivariate, and one having two or more variables is said to be multivariate. In a polynomial, the variables correspond to the base symbols themselves stripped of coefficients and any powers or products.
The variables in a polynomial can be extracted using the Mathematica command Variables.
The field of all rational and irrational numbers is called the real numbers, or simply the "reals," and denoted . The set of real numbers is also called the continuum, denoted . The set of reals is called Reals in Mathematica, and a number can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real.
The real numbers can be extended with the addition of the imaginary number i, equal to (-1)^1/2. Numbers of the form x+iy, where and are both real, are called complex numbers, which also form a field. Another extension which includes both the real numbers and the infinite ordinal numbers of Georg Cantor is the surreal numbers.
Friday, May 15, 2009
polynomial
In mathematics, a polynomial is a finite length expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number.
Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.
A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent that is a non-negative integer (also known as a natural number). The exponent on a variable in a term is equal to the degree of that variable in that term. Since x = x1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and complex numbers.
For example,
is a term. The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.
A polynomial is a sum of terms. For example, the following is a polynomial:
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.
When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In this example, the polynomial has degree two.
Alternative forms
An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance,
is a polynomial because it can be worked out to x3 + 3x2 + 3x + 1. Similarly,
is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to and is just a constant. The coefficient of this term is therefore . For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i)x3; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.
Division by an expression containing a variable is not generally allowed in polynomials.
Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.
Polynomial functions
A polynomial function is a function defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies
for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by
is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in
Polynomial functions are an important class of smooth functions.
Polynomial equations
A polynomial equation is an equation in which a polynomial is set equal to another polynomial.
is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x2–y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality.
A sum of polynomials is a polynomial.
A product of polynomials is a polynomial
The derivative of a polynomial function is a polynomial function
Any primitive or antiderivative of a polynomial function is a polynomial function
Polynomials serve to approximate other functions, such as sine, cosine, and exponential.
All polynomials have an expanded form, in which the distributive law has been used to remove all brackets. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear complex polynomials. For example, the polynomial
is the expanded form of the polynomial
,
which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
Every polynomial in one variable is equivalent to a polynomial with the form
This form is sometimes taken as the definition of a polynomial in one variable.
Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme
In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.
A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. This important subject is studied extensively in the area of mathematics known as linear algebra. Overdetermined systems are common in practical applications.
Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.
A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent that is a non-negative integer (also known as a natural number). The exponent on a variable in a term is equal to the degree of that variable in that term. Since x = x1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and complex numbers.
For example,
is a term. The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.
A polynomial is a sum of terms. For example, the following is a polynomial:
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.
When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In this example, the polynomial has degree two.
Alternative forms
An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance,
is a polynomial because it can be worked out to x3 + 3x2 + 3x + 1. Similarly,
is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to and is just a constant. The coefficient of this term is therefore . For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i)x3; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.
Division by an expression containing a variable is not generally allowed in polynomials.
Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.
Polynomial functions
A polynomial function is a function defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies
for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by
is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in
Polynomial functions are an important class of smooth functions.
Polynomial equations
A polynomial equation is an equation in which a polynomial is set equal to another polynomial.
is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x2–y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality.
A sum of polynomials is a polynomial.
A product of polynomials is a polynomial
The derivative of a polynomial function is a polynomial function
Any primitive or antiderivative of a polynomial function is a polynomial function
Polynomials serve to approximate other functions, such as sine, cosine, and exponential.
All polynomials have an expanded form, in which the distributive law has been used to remove all brackets. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear complex polynomials. For example, the polynomial
is the expanded form of the polynomial
,
which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
Every polynomial in one variable is equivalent to a polynomial with the form
This form is sometimes taken as the definition of a polynomial in one variable.
Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme
In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.
A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. This important subject is studied extensively in the area of mathematics known as linear algebra. Overdetermined systems are common in practical applications.
Tuesday, May 12, 2009
WELCOME
Welcome all of you .
i am strat this blog on the topic of mathematics ,the most hard subject. but i try form this blog to change the mantility of student and people .math is not a hard subject it is a intersted subject . it is a game and enjoy with me the game of mathematics.
Here i post my artical related to all the topic of mathematic . i divide my matterial accounding to the class and level .so i though you all are enjoying the fun of math with me .
thanks all of you for your valuble time.
Grate to see you here.
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