Algebraic vs. Polynomial — What's the Difference?
By Tayyaba Rehman & Fiza Rafique — Updated on May 16, 2024
Algebraic refers to expressions, equations, and structures involving variables and operations, polynomial specifically denotes a type of algebraic expression consisting of variables and coefficients combined using addition, subtraction, multiplication.
Difference Between Algebraic and Polynomial
Table of Contents
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Key Differences
Algebraic refers to anything related to algebra, including various structures, expressions, equations, and operations involving variables. Polynomial, on the other hand, is a specific type of algebraic expression that consists of terms with non-negative integer exponents on the variables.
An algebraic expression can encompass a wide range of mathematical forms, including polynomials, rational expressions, and more complex constructions. Polynomials are more restricted in form, composed of a sum of terms, each being a product of a constant coefficient and a variable raised to a non-negative integer power.
In algebra, operations such as addition, subtraction, multiplication, and division are commonly used, often involving variables and constants. Polynomials also utilize these operations, but they do not involve division by a variable; every term in a polynomial must have variables with non-negative integer exponents.
Algebraic equations can be linear, quadratic, or involve higher degrees and complex structures. Polynomial equations specifically involve polynomial expressions set equal to zero, and they can also be linear, quadratic, cubic, etc., depending on the highest power of the variable present.
The field of algebra includes various sub-disciplines like linear algebra, abstract algebra, and algebraic geometry, encompassing a broad range of mathematical theories and applications. Polynomials are a fundamental component within these fields but represent only a subset of the broader algebraic landscape.
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Comparison Chart
Definition
Involving variables and operations in expressions or equations
Specific type of expression with variables and coefficients
Scope
Broad, including many types of expressions and equations
Narrow, only includes sums of terms with non-negative integer exponents
Operations
Addition, subtraction, multiplication, division
Addition, subtraction, multiplication, no division by variables
Equation Types
Linear, quadratic, higher degree, complex structures
Linear, quadratic, cubic, etc., specific to polynomial forms
Sub-disciplines
Linear algebra, abstract algebra, algebraic geometry
Fundamental in these fields but not a separate sub-discipline
Compare with Definitions
Algebraic
Pertaining to algebraic numbers or functions.
An algebraic number is a root of a non-zero polynomial equation with rational coefficients.
Polynomial
An algebraic expression that includes terms added or subtracted together.
Polynomials can be used to model real-world situations like projectile motion.
Algebraic
Relating to algebra, involving variables and operations.
Solving algebraic equations requires knowledge of various mathematical operations.
Polynomial
A mathematical expression consisting of variables, coefficients, and non-negative integer exponents.
The polynomial 3x^2 + 2x + 1 is quadratic in nature.
Algebraic
Describing relationships using algebraic methods.
The algebraic formula represents the relationship between variables.
Polynomial
Formed by the sum of multiple terms, each being a product of a constant and a variable raised to an exponent.
In the polynomial 4x^3 + x - 7, each term represents a part of the expression.
Algebraic
Of, relating to, or designating algebra.
Polynomial
Capable of being classified by their degree, such as linear, quadratic, cubic, etc.
The polynomial x^2 - 4x + 4 is quadratic because the highest exponent is two.
Algebraic
Designating an expression, equation, or function in which only numbers, letters, and arithmetic operations are contained or used.
Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7.
Algebraic
Indicating or restricted to a finite number of operations involving algebra.
Polynomial
Of, relating to, or consisting of more than two names or terms.
Algebraic
Of, or relating to, algebra.
Polynomial
A taxonomic designation consisting of more than two terms.
Algebraic
Containing only numbers, letters, and arithmetic operators.
Polynomial
An algebraic expression consisting of one or more summed terms, each term consisting of a constant multiplier and one or more variables raised to nonnegative integral powers. For example, x2 - 5x + 6 and 2p3q + y are polynomials. Also called multinomial.
Algebraic
Which is a root of some polynomial whose coefficients are rational.
Polynomial
An expression of two or more terms.
Algebraic
Whose every element is a root of some polynomial whose coefficients are rational.
Polynomial
An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as . Category:en:Polynomials
Algebraic
Describing squares by file (referred to in intrinsic order rather than by the piece starting on that file) and rank, both with reference to a fixed point rather than a player-dependent perspective.
Polynomial
(taxonomy) A taxonomic designation (such as of a subspecies) consisting of more than two terms.
Algebraic
Of or pertaining to algebra; using algebra; according to the laws of algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings; algebraic geometry.
Polynomial
(algebra) Able to be described or limited by a polynomial.
Algebraic
Progressing by constant multiplicatory factors; - of a series of numbers. Contrasted to arithmetical.
Polynomial
(taxonomy) of a polynomial name or entity
Algebraic
Of or relating to algebra;
Algebraic geometry
Polynomial
An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.
Algebraic
Involving algebraic structures such as groups, rings, and fields.
Group theory is a branch of mathematics studying algebraic structures.
Polynomial
Containing many names or terms; multinominal; as, the polynomial theorem.
Polynomial
Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.
Polynomial
A mathematical expression that is the sum of a number of terms
Polynomial
Having the character of a polynomial;
A polynomial expression
Polynomial
Defined by the highest power of its variable, known as its degree.
A cubic polynomial has a degree of three.
Common Curiosities
Can all algebraic equations be considered polynomial equations?
No, not all algebraic equations are polynomial equations; some may involve more complex structures or operations.
What does algebraic mean?
It refers to anything related to algebra, including expressions, equations, and operations involving variables.
What is a polynomial?
A polynomial is an algebraic expression consisting of terms with non-negative integer exponents on the variables.
What operations are used in polynomials?
Polynomials use addition, subtraction, and multiplication but do not involve division by variables.
Is algebraic geometry related to polynomials?
Yes, algebraic geometry often studies solutions to polynomial equations and their properties.
What is the degree of a polynomial?
The degree is the highest exponent of the variable in the polynomial.
Are all algebraic numbers roots of polynomials?
Yes, algebraic numbers are defined as roots of non-zero polynomial equations with rational coefficients.
Do polynomials have applications in real life?
Yes, polynomials are used in various fields such as physics, engineering, and economics for modeling and problem-solving.
How do algebraic expressions differ from polynomials?
Algebraic expressions can include a wider range of operations and forms, while polynomials are specifically sums of terms with non-negative integer exponents.
What are some examples of algebraic structures?
Examples include groups, rings, and fields in abstract algebra.
What distinguishes a linear polynomial from a non-linear one?
A linear polynomial has a degree of 1, while non-linear polynomials have degrees greater than 1.
How is abstract algebra different from studying polynomials?
Abstract algebra focuses on algebraic structures like groups and rings, whereas studying polynomials focuses on specific types of algebraic expressions.
Are quadratic equations always polynomials?
Yes, quadratic equations are polynomial equations with a degree of 2.
Can algebraic expressions include roots and exponents other than polynomials?
Yes, algebraic expressions can include roots, fractional exponents, and other forms beyond polynomials.
Can polynomials have negative exponents?
No, polynomials have terms with non-negative integer exponents only.
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Written by
Tayyaba RehmanTayyaba Rehman is a distinguished writer, currently serving as a primary contributor to askdifference.com. As a researcher in semantics and etymology, Tayyaba's passion for the complexity of languages and their distinctions has found a perfect home on the platform. Tayyaba delves into the intricacies of language, distinguishing between commonly confused words and phrases, thereby providing clarity for readers worldwide.
Co-written by
Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.
