What is a Polynomial?
A polynomial is an algebraic expression made up of terms, where each term consists of a variable raised to a non-negative integer power multiplied by a coefficient. The word "polynomial" comes from the Greek words "poly" (meaning many) and "nomial" (meaning terms).
For example, 3x² + 5x - 7 is a polynomial with three terms. The first term is 3x², the second term is 5x, and the third term is -7. Each term has a coefficient (3, 5, and -7 respectively) and the variable x is raised to different powers (2, 1, and 0).
Standard Form of a Polynomial
A polynomial in one variable x is generally written in the form:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀
Where:
- aₙ, aₙ₋₁, ..., a₁, a₀ are real numbers called coefficients
- aₙ ≠ 0 (the leading coefficient cannot be zero)
- n is a non-negative integer
- x is the variable
Key Terminology
1. Terms of a Polynomial
Each part of a polynomial separated by plus or minus signs is called a term. In the polynomial 4x³ - 2x² + 7x - 5, there are four terms: 4x³, -2x², 7x, and -5.
2. Coefficients
The numerical factor in each term is called the coefficient. In the term 6x², the coefficient is 6. If no number is written, the coefficient is understood to be 1 (for example, x² has a coefficient of 1).
3. Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial. For example, the polynomial 5x⁴ + 3x² - 8x + 1 has degree 4 because the highest power of x is 4.
Important: The degree of a non-zero constant polynomial (like 5 or -3) is 0, because it can be written as 5x⁰. The zero polynomial (0) has no defined degree.
4. Leading Coefficient
The coefficient of the term with the highest degree is called the leading coefficient. In 3x⁴ + 2x² - 7, the leading coefficient is 3.
5. Constant Term
The term without any variable is called the constant term. In 2x³ + 5x - 9, the constant term is -9.
Types of Polynomials Based on Number of Terms
Monomial
A polynomial with exactly one term is called a monomial.
Examples: 5x, -3x², 7, 2x⁴
Binomial
A polynomial with exactly two terms is called a binomial.
Examples: x + 5, 3x² - 2x, 7y - 4
Trinomial
A polynomial with exactly three terms is called a trinomial.
Examples: x² + 5x + 6, 2a² - 3a + 1, x⁴ - x² + 3
Polynomial
When there are more than three terms, we simply call it a polynomial.
Example: x⁴ + 3x³ - 2x² + 5x - 7
Types of Polynomials Based on Degree
Linear Polynomial (Degree 1)
A polynomial of degree 1 is called a linear polynomial. Its general form is ax + b, where a ≠ 0.
Examples: 2x + 3, 5x - 7, -x + 4
Quadratic Polynomial (Degree 2)
A polynomial of degree 2 is called a quadratic polynomial. Its general form is ax² + bx + c, where a ≠ 0.
Examples: x² + 5x + 6, 3x² - 2x + 1, -2x² + 7
Cubic Polynomial (Degree 3)
A polynomial of degree 3 is called a cubic polynomial. Its general form is ax³ + bx² + cx + d, where a ≠ 0.
Examples: x³ + 2x² - x + 5, 2x³ - 3x + 1
Bi-quadratic Polynomial (Degree 4)
A polynomial of degree 4 is called a bi-quadratic polynomial.
Example: x⁴ + 3x² - 5
Important Points to Remember
- The exponents of variables in a polynomial must be non-negative integers (0, 1, 2, 3, ...)
- Expressions like x⁻¹, x^(1/2), or √x are NOT polynomials
- A polynomial can have only one variable or multiple variables
- The degree of the zero polynomial is not defined
- Every constant (except 0) is a polynomial of degree 0
Common Misconceptions
Misconception 1: Negative exponents create polynomials
Wrong: x⁻² + 3x is a polynomial
Correct: Polynomials can only have non-negative integer exponents. x⁻² + 3x is NOT
a polynomial.
Misconception 2: Fractional exponents are allowed
Wrong: x^(1/2) + 2x is a polynomial
Correct: x^(1/2) means √x, which is not allowed in polynomials.
Misconception 3: The degree is the number of terms
Wrong: The polynomial x² + 3x - 5 has degree 3
Correct: The degree is 2 (highest power), not 3 (number of terms).
Real-World Applications
Polynomials are used extensively in:
- Physics: Describing motion (position, velocity, acceleration)
- Economics: Modeling cost and revenue functions
- Engineering: Design calculations and curve fitting
- Computer Science: Algorithm analysis and graphics
Summary
Polynomials are algebraic expressions with terms containing variables raised to non-negative integer powers. Understanding polynomial terminology—terms, coefficients, degree, and types—is fundamental to working with these expressions. In the next chapters, you'll learn how to perform operations on polynomials and factor them.