Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
Key Formula
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
LaTeX: P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
| Symbol | Meaning | Unit |
|---|---|---|
| P(x) | Polynomial function of x | dimensionless |
| aₙ | Leading coefficient (must be non-zero) | dimensionless |
| n | Degree of the polynomial (highest exponent) | dimensionless |
| a₀ | Constant term (value when x = 0) | dimensionless |
Worked Example
Problem
Evaluate the polynomial P(x) = 2x³ − 3x² + x − 5 at x = 2 using Horner's method.
Solution
Horner's method rewrites P(x) = ((2x − 3)x + 1)x − 5 Step 1: Start with the leading coefficient: 2 Step 2: Multiply by x = 2 and add next coefficient: 2×2 + (−3) = 4 − 3 = 1 Step 3: Multiply by 2 and add next coefficient: 1×2 + 1 = 3 Step 4: Multiply by 2 and add constant: 3×2 + (−5) = 6 − 5 = 1 Verify directly: 2(8) − 3(4) + 2 − 5 = 16 − 12 + 2 − 5 = 1 ✓
Answer
P(2) = 1
Classification of Polynomials by Degree
| Degree | Name | Standard Form | Example |
|---|---|---|---|
| 0 | Constant | a₀ | 7 |
| 1 | Linear | a₁x + a₀ | 4x − 3 |
| 2 | Quadratic | a₂x² + a₁x + a₀ | x² − 2x + 1 |
| 3 | Cubic | a₃x³ + a₂x² + a₁x + a₀ | 2x³ + x − 4 |
| 4 | Quartic | a₄x⁴ + … + a₀ | x⁴ − 5x² + 4 |
| 5 | Quintic | a₅x⁵ + … + a₀ | 3x⁵ − x³ + 2x |
Interactive Tools
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Related Terms
Algebraic Expression
An algebraic expression is a combination of variables, constants, and arithmetic operations (addition, subtraction, multiplication, division, exponentiation) that represents a mathematical quantity without an equality sign. Unlike equations, expressions do not assert that two things are equal — they simply describe a value that depends on the variables present. Examples include 3x + 5, 2a² − 4b, and (x + y)/2.
Factoring
Factoring (or factorisation) in algebra is the process of rewriting an algebraic expression as a product of simpler expressions called factors, reversing the process of expansion. For example, x² − 5x + 6 can be factored as (x − 2)(x − 3). Factoring is essential for solving polynomial equations, simplifying rational expressions, and finding roots, and it is a core skill that underpins much of higher mathematics.
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and it can have two, one, or no real solutions depending on the value of the discriminant (b² − 4ac). Quadratic equations model projectile motion, area problems, and many optimisation scenarios in physics and engineering.
From Greek "polynomion", combining "poly" (many) and "nomos" (part or portion), later Latinised. The term was introduced into mathematics in the 16th century. The systematic theory of polynomials was developed by European mathematicians including Viète, Descartes, and later Gauss.