Polynomials: Definition & Key Terms

Definition: Polynomial in one variable, terms, coefficients, degree.

This section introduces the fundamental concepts of polynomials in a single variable, including the definition, key components like terms, coefficients, and degree. Understanding these concepts is crucial for further study in algebra and calculus.

A polynomial in one variable (usually denoted by *x*) is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, and non-negative integer exponents of the variable.

Let’s break down the key terms:

Terms: The individual parts of a polynomial separated by addition or subtraction signs. Each term consists of a coefficient and a variable raised to a non-negative integer power.
Coefficients: The numerical values multiplying the variables in each term.
Degree: The highest power of the variable in the polynomial.

Formulae

The general form of a polynomial in one variable, *x*, can be written as:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_2 x^2 + a_1 x + a_0$

$a_n, a_{n-1}, …, a_2, a_1, a_0$ are the coefficients.
$n$ is a non-negative integer, and represents the degree of the polynomial.
The term $a_0$ is the constant term (the coefficient of $x^0$).

Examples

Here are some examples to illustrate the concepts:

Example-1:

Consider the polynomial: $3x^2 + 2x – 1$

Terms: $3x^2$, $2x$, and $-1$
Coefficients: 3, 2, and -1
Degree: 2 (because the highest power of *x* is 2)

Example-2:

Consider the polynomial: $5x^3 – 7x + 4$

Terms: $5x^3$, $-7x$, and $4$
Coefficients: 5, -7, and 4
Degree: 3 (because the highest power of *x* is 3)

Common mistakes by students

Incorrectly identifying terms: Forgetting to include the sign (+ or -) in front of a term. Example: In $2x^2 – 3x + 1$, a common mistake is to list the terms as $2x^2$, $3x$, and $1$ instead of $2x^2$, $-3x$, and $1$.
Misunderstanding the degree: Incorrectly stating the degree by looking at only the first term. The degree is the highest power of the variable in *any* term.
Confusing coefficients and exponents: Mixing up the coefficient and the power of the variable. For example, in $4x^3$, confusing the coefficient 4 with the exponent 3.
Considering negative exponents or fractional powers: Only non-negative integers are permitted in the exponents of polynomials. Expressions like $x^{-1}$ or $x^{1/2}$ are not polynomials.

Real Life Application

Polynomials are used extensively in real-world applications, including:

Modeling curves: Used to create smooth curves in computer graphics and engineering design.
Calculating trajectories: Used in physics to model the path of projectiles.
Economic forecasting: Used to model trends in markets and predict future values.
Engineering: Used in structural analysis and other fields.

Fun Fact

The word “polynomial” comes from the Greek words “poly” (meaning “many”) and “nomial” (meaning “term”). So a polynomial is literally an expression with many terms!