Polynomials: Definition, Classification, And Applications

Polynomial

An algebraic expression that is the sum of one or more terms (or monomials)

A polynomial is an algebraic expression that consists of one or more terms, each of which is a product of a numerical coefficient and one or more variables raised to non-negative integer powers. For example, the expression 3x^2 + 2x – 7 is a polynomial consisting of three terms. The highest degree of a polynomial is the largest exponent of its variable(s).

Polynomials can be classified according to their degree. A polynomial of degree zero is a constant, such as 7 or -9. A polynomial of degree one is a linear function of form ax + b, where a and b are constants. A polynomial of degree two is a quadratic function of form ax^2 + bx + c, where a, b, and c are constants. A polynomial of degree three is a cubic function of form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. Higher degree polynomials are named according to their degree, e.g., polynomial of degree four is a quartic polynomial.

Polynomials can be added, subtracted, multiplied, and divided using various techniques. Polynomial long division and synthetic division are two methods of dividing one polynomial by another. Factoring is a technique used to simplify an expression by writing it as a product of polynomials, such as through factoring to (x + 2)(x – 3) from x^2 – x – 6.

Polynomials are used to model many real-world situations, such as calculating areas and volumes of geometric shapes, predicting the behavior of physical systems, and analyzing the performance of algorithms. Polynomials are also widely used in Computer Science, Finance, Engineering, and many other fields.

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