Types of Polynomials: Classification

Types of Polynomials: Linear, Quadratic, Cubic; Monomials, Binomials, Trinomials

Polynomials are algebraic expressions consisting of variables and coefficients, that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are classified based on two main criteria: the degree and the number of terms.

Degree: The highest power of the variable in the polynomial.

Terms: The individual parts of a polynomial separated by addition or subtraction signs.

Formulae

Here’s a table summarizing the degree and general forms of common polynomial types:

Type	Degree	General Form
Linear	1	$ax + b$
Quadratic	2	$ax^2 + bx + c$
Cubic	3	$ax^3 + bx^2 + cx + d$
Monomial	Varies	$ax^n$ (n is a non-negative integer)
Binomial	Varies	$ax^n + bx^m$ (n and m are non-negative integers)
Trinomial	Varies	$ax^n + bx^m + cx^p$ (n, m, and p are non-negative integers)

Examples

Example-1: Classify the polynomial $3x^2 + 2x – 1$.

It is a quadratic polynomial (degree 2) and a trinomial (3 terms).

Example-2: Classify the polynomial $5x^3$.

It is a cubic polynomial (degree 3) and a monomial (1 term).

Common mistakes by students

Incorrect Degree Identification: Students often misidentify the degree of a polynomial. For instance, they might miss the highest exponent. Always look for the largest exponent to determine the degree.
Confusing Terms and Degree: Mixing up the number of terms with the degree. Remember terms are separated by + or – signs, while the degree is the highest power.
Simplifying Incorrectly: Students sometimes attempt to combine terms that are not like terms (e.g., $x^2$ and $x$). Remember, only like terms can be added or subtracted.
Failing to Recognize Special Forms: Difficulty identifying a linear form or quadratic expression when terms are not in their standard form.

Real Life Application

Polynomials are used extensively in various fields:

Physics: Modeling projectile motion (quadratic equations)
Engineering: Designing curves and surfaces, analyzing the behavior of structures (cubic equations)
Economics: Modeling cost, revenue, and profit functions (quadratic equations)
Computer Graphics: Creating smooth curves and surfaces (polynomial interpolation)

Fun Fact

The term “polynomial” comes from the Greek words “poly” (meaning many) and “nomial” (meaning name). This makes sense because polynomials are algebraic expressions that can have many terms. The degree can also be thought of as the “order” of the polynomial.