Polynomial

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Polynomial refers to a mathematical expression consisting of variables, coefficients, and operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials are a fundamental part of algebra and form the basis for more advanced topics in mathematics, such as calculus and algebraic geometry.

Definition

A polynomial in a single variable x can be written in the form: \[a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0\] where:

• n is a non-negative integer,
• a_n, a_{n-1}, ..., a_1, a_0 are coefficients, which are numbers from a given set, often the set of real numbers or complex numbers,
• x is the variable.

The highest power of x that appears in the polynomial is called the degree of the polynomial. If the highest degree of the polynomial is n, then the polynomial is said to be of nth degree. The coefficient of the term with the highest degree is called the leading coefficient.

Types of Polynomials

Polynomials can be classified based on their number of terms:

• A monomial has one term.
• A binomial has two terms.
• A trinomial has three terms.
• A polynomial with more than three terms is usually just called a polynomial.

Polynomials can also be classified based on their degree:

• A polynomial of degree 0 is a constant function.
• A polynomial of degree 1 is a linear function.
• A polynomial of degree 2 is a quadratic function.
• A polynomial of degree 3 is a cubic function.
• And so on for higher degrees.

Operations on Polynomials

Polynomials can be added, subtracted, multiplied, and divided (except by zero) under certain conditions. The result of these operations is another polynomial.

Addition and Subtraction

To add or subtract polynomials, combine like terms, which are terms that have the same variable raised to the same power.

Multiplication

To multiply polynomials, use the distributive property to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.

Division

Polynomial division can be performed using long division or synthetic division, but it may result in a remainder. When dividing polynomials, the result may not always be another polynomial.

Polynomial Functions

A polynomial function is a function that is defined by a polynomial. For example, the function f(x) = x^2 + 3x - 5 is a polynomial function of degree 2. Polynomial functions are continuous and smooth, without breaks, jumps, or sharp corners.

Roots of a Polynomial

The roots (or zeros) of a polynomial are the values of x for which the polynomial equals zero. Finding the roots of a polynomial is a central task in algebra. The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex root, and a polynomial of degree n has exactly n roots, counting multiplicities.

Applications

Polynomials are used in a wide range of areas in both pure and applied mathematics. They are used to construct polynomial equations which model a wide variety of phenomena, including physical, biological, and economic systems. Polynomials also play a crucial role in numerical analysis, engineering, and the sciences.

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