Equation vs. Polynomial — What's the Difference?
By Urooj Arif & Maham Liaqat — Updated on April 1, 2024
An equation is a mathematical statement asserting the equality of two expressions, often containing variables and constants. A polynomial is a specific type of mathematical expression composed of variables, coefficients, and exponents arranged in a sum.
Difference Between Equation and Polynomial
Table of Contents
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Key Differences
An equation functions as a statement that two expressions are equal, indicated by the "=" sign. It can encompass a wide variety of forms, including linear, quadratic, and polynomial equations, setting the stage for solving unknown values. Whereas a polynomial is a particular form of expression, not necessarily equating to anything, characterized by its components: variables raised to whole number exponents and their coefficients.
Equations can be used to represent relationships between quantities and to solve problems by finding the values of unknown variables that make the equation true. This process involves manipulation and transformation of the equation while maintaining its balance. On the other hand, polynomials serve as the building blocks of polynomial equations when set equal to another expression, typically zero or another polynomial, but stand alone as expressions defining a sum of terms.
The complexity of an equation is not limited by the degree of its variables, as it can include a variety of operations beyond addition and subtraction, such as logarithmic or trigonometric functions. In contrast, the complexity of a polynomial is specifically determined by its highest degree (the largest exponent of its variables) and the number of terms it contains, sticking to operations of addition, subtraction, and multiplication by constants.
While equations can represent a wide range of mathematical relationships and are used in fields as diverse as physics and economics to model phenomena, polynomials are particularly valued in algebra and calculus for their properties and behavior. They play a crucial role in numerical analysis, mathematical modeling, and solving polynomial equations.
Equations and polynomials serve different purposes in mathematics: equations are statements of equality that can be solved or used to model relationships, while polynomials are expressions that can form one or both sides of an equation but are primarily characterized by their structure and the operations that define them.
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Comparison Chart
Definition
A statement that asserts the equality of two expressions.
An expression consisting of variables, coefficients, and non-negative integer exponents.
Components
Can include variables, constants, coefficients, and a variety of mathematical operations.
Comprised of terms that are variables raised to whole number exponents and multiplied by coefficients.
Purpose
To represent a relationship between quantities or to find the values of unknowns.
To serve as an expression or part of an equation, particularly useful in algebraic operations.
Complexity
Determined by the variety of operations (addition, subtraction, multiplication, division, etc.) and the level of the mathematical functions involved.
Determined by the degree (highest exponent of the variable) and the number of terms.
Compare with Definitions
Equation
Describes physical laws in sciences.
Newton's second law is often written as the equation F=ma.
Polynomial
Used in polynomial functions to describe a wide range of mathematical behavior.
The graph of the polynomial shows its roots.
Equation
Can represent conditions in word problems.
The problem can be modeled by a quadratic equation.
Polynomial
Plays a key role in calculus and algebra.
The derivative of a polynomial is another polynomial.
Equation
Serves as the foundation for solving algebraic problems.
Solving the equation requires isolating the variable.
Polynomial
An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
The polynomial −4x+7 is cubic.
Equation
In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.Solving an equation containing variables consists of determining which values of the variables make the equality true.
Polynomial
Defined by its degree and the number of its terms.
A quadratic polynomial has a degree of 2.
Equation
The act or process of equating or of being equated.
Polynomial
Can be simplified or expanded through algebraic operations.
Expanding the polynomial results in a simpler expression.
Equation
The state of being equal.
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.
Equation
(Mathematics) A statement asserting the equality of two expressions, usually written as a linear array of symbols that are separated into left and right sides and joined by an equal sign.
Polynomial
Of, relating to, or consisting of more than two names or terms.
Equation
(Chemistry) A representation of a chemical reaction, usually written as a linear array in which the symbols and quantities of the reactants are separated from those of the products by an arrow or a set of opposing arrows.
Polynomial
A taxonomic designation consisting of more than two terms.
Equation
A complex of variable elements or factors
"The world was full of equations ... there must be an answer for everything, if only you knew how to set forth the questions" (Anne Tyler).
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.
Equation
The act or process of equating two or more things, or the state of those things being equal (that is, identical).
We need to bring the balance of power into equation
Polynomial
An expression of two or more terms.
Equation
(mathematics) An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
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
Equation
(astronomy) A small correction to observed values to remove the effects of systematic errors in an observation.
Polynomial
(taxonomy) A taxonomic designation (such as of a subspecies) consisting of more than two terms.
Equation
A making equal; equal division; equality; equilibrium.
Again the golden day resumed its right,And ruled in just equation with the night.
Polynomial
(algebra) Able to be described or limited by a polynomial.
Equation
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
Polynomial
(taxonomy) of a polynomial name or entity
Equation
A quantity to be applied in computing the mean place or other element of a celestial body; that is, any one of the several quantities to be added to, or taken from, its position as calculated on the hypothesis of a mean uniform motion, in order to find its true position as resulting from its actual and unequal motion.
Polynomial
An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.
Equation
A mathematical statement that two expressions are equal
Polynomial
Containing many names or terms; multinominal; as, the polynomial theorem.
Equation
A state of being essentially equal or equivalent; equally balanced;
On a par with the best
Polynomial
Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.
Equation
The act of regarding as equal
Polynomial
A mathematical expression that is the sum of a number of terms
Polynomial
Having the character of a polynomial;
A polynomial expression
Common Curiosities
Can a polynomial be an equation?
A polynomial becomes an equation when it is set equal to something else, typically zero or another polynomial.
How does the degree of a polynomial affect its complexity?
The degree, or the highest exponent of its variable, largely determines a polynomial's complexity and the shape of its graph.
Can equations only contain polynomials?
No, equations can contain a wide variety of mathematical expressions, including but not limited to polynomials.
Why are polynomials important in mathematics?
Polynomials are fundamental in algebra and calculus, useful for describing a wide range of mathematical phenomena and solving equations.
How do you solve a polynomial equation?
Solving a polynomial equation often involves finding the values of the variable that make the polynomial equal to zero or another value.
What makes a polynomial linear?
A polynomial is linear if it has a degree of one, meaning its highest exponent on the variable is one.
What is the role of variables in equations and polynomials?
Variables represent unknown values in equations and are used to indicate quantities that can change in polynomials.
What distinguishes an equation from a polynomial?
An equation asserts equality between two expressions, while a polynomial is a type of expression itself.
Are all equations linear?
No, equations can be linear, quadratic, polynomial, or involve more complex relationships between variables.
Why are equations important in science?
Equations are essential for modeling physical laws, describing scientific phenomena, and making predictions.
How are equations used in real life?
Equations model relationships between quantities, used in fields like engineering, physics, finance, and many others to solve real-world problems.
What is the simplest form of a polynomial?
The simplest polynomial is a monomial, consisting of only one term.
How are polynomials added or subtracted?
Polynomials are added or subtracted by combining like terms, which are terms with the same variables raised to the same power.
What is a zero of a polynomial?
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero, indicating where its graph intersects the x-axis.
Can polynomials have negative exponents?
No, polynomials only include non-negative integer exponents.
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Written by
Urooj ArifUrooj is a skilled content writer at Ask Difference, known for her exceptional ability to simplify complex topics into engaging and informative content. With a passion for research and a flair for clear, concise writing, she consistently delivers articles that resonate with our diverse audience.
Co-written by
Maham Liaqat
