Introduction to Polynomials

DEFINITION AND FORM

A polynomial is an algebraic expression composed of the sum or difference of terms, in which the unknown x (or another variable) appears with non-negative integer exponents, multiplied by coefficients from some set of numbers (most commonly ℝ - real numbers). The general form of a polynomial of degree n is:

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀,
where a₀, a₁, …, aₙ ∈ ℝ, aₙ ≠ 0, and n ∈ ℕ₀ (n is a non-negative integer).

Each part of the form aₖxᵏ is called a term of the polynomial, where:

• aₖ is the coefficient,
• xᵏ is a power of the variable,
• k is a natural number or 0 (a non-negative integer).

DEGREE OF A POLYNOMIAL

The degree (notation deg P) is the highest exponent at which the coefficient is different from zero.
For example:
P(x) = 4x³ – 2x + 7 → deg P = 3.

TYPES OF POLYNOMIALS BASED ON DEGREE

• CONSTANT POLYNOMIAL: degree 0, e.g., P(x) = 5 (can be written as 5x⁰)
• LINEAR POLYNOMIAL: degree 1, e.g., P(x) = 2x – 1
• QUADRATIC POLYNOMIAL: degree 2, e.g., P(x) = x² + 3x + 2
• CUBIC POLYNOMIAL: degree 3, e.g., P(x) = x³ – x
• POLYNOMIALS OF HIGHER DEGREES: n > 3

BASIC OPERATIONS WITH POLYNOMIALS

• ADDITION AND SUBTRACTION – Combine like terms (term by term).
• MULTIPLICATION – Multiply terms using distributivity.
• DIVISION – By a process similar to long division of numbers.
• DIFFERENTIATION – In higher mathematics (e.g., P′(x), the derivative).
• EVALUATING THE POLYNOMIAL – Calculating P(x₀) for a specific x₀.

EXAMPLE OF A POLYNOMIAL

Let the given polynomial be:
P(x) = 2x⁴ – 5x² + 3x – 7

• Degree: 4
• Leading coefficient: 2
• Constant term: –7
• Value at x=1: P(1) = 2*(1)⁴ – 5*(1)² + 3*(1) – 7 = 2 – 5 + 3 – 7 = –7

CONCLUSION

Polynomials are fundamental algebraic expressions that allow for the writing and handling of numerous mathematical functions and equations. Their structure is clear, operations with them are systematic, and therefore they form the basis for solving equations, modeling, and mathematical analysis in many chapters of mathematics.