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Polynomial expressions are foundational mathematical constructs involving variables, coefficients, and arithmetic operations. They form the basis of polynomial equations and functions, impacting disciplines like physics and economics. The term 'polynomial' has Greek and Latin roots, reflecting its historical importance. Variables play a dual role as placeholders or function inputs, while the structure of polynomials allows for equivalent expressions through algebraic laws.
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Polynomials are mathematical expressions composed of variables and coefficients, combined using operations such as addition, subtraction, multiplication, and exponentiation
Polynomials in One Variable
Polynomials in one variable, such as x² - 4x + 7, can be simple or involve multiple variables
Polynomials in Multiple Variables
Polynomials in multiple variables, such as x³ + 2xyz² - yz + 1, have practical applications in various disciplines
Polynomials are used to model problems in fields such as physics and economics, and are essential in constructing polynomial functions
The term "polynomial" comes from the Greek prefix "poly-" meaning "many," and the Latin root "nomen," meaning "name" or "term."
Polynomials have been a part of mathematical language since the 17th century, demonstrating their enduring importance in mathematical discourse
Variables, such as x, serve as placeholders or inputs in polynomial expressions and functions
Dual Notation for Polynomials
The notation P(x) denotes a polynomial with the indeterminate x, while P alone names the polynomial, illustrating the dual notation for polynomials
Notation for Polynomial Functions
The notation a → P(a) represents the mapping of a polynomial P to an input a, applicable over any mathematical ring
Polynomial expressions are composed of constants, variables, and operations of addition, multiplication, and exponentiation
Polynomial expressions that can be transformed into each other using the commutative, associative, and distributive laws are considered equivalent
A polynomial can always be written in standard form as a sum of terms with coefficients and variables raised to non-negative integer powers, expressed using summation notation
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A simple polynomial in one variable might look like x² - 4x + 7, while a more complex one could include terms like ______ + 2xyz² - yz + 1.
x³
01
Polynomials are not just theoretical; they have real-world uses in fields such as ______, economics, and many others.
physics
02
Origin of 'poly-' in 'polynomial'
Greek prefix 'poly-' means 'many'.
