Polynomial Evaluator

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication with non-negative integer exponents:

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Where:

• aₙ, aₙ₋₁, …, a₁, a₀ are the coefficients (the numbers multiplying each power of x)
• n is the degree of the polynomial (the highest power of x)
• x is the variable

How to enter coefficients: Enter the coefficients separated by commas, from the highest degree down to the constant term (the number with no x).

Example: For 3x³ + 2x² - 5x + 7, enter: 3, 2, -5, 7 If a term is missing (e.g., x³ + 4), you must include zeros for the missing powers: 1, 0, 0, 4

Evaluation method — Horner’s Method: The calculator uses Horner’s method, which rewrites the polynomial to minimize multiplications: P(x) = (...((aₙx + aₙ₋₁)x + aₙ₋₂)x + ...) + a₀ This is both faster and more numerically stable than computing each power of x separately.

Practical Example: Evaluate P(x) = 3x³ + 2x² - 5x + 7 at x = 2: Using Horner’s method: ((3 x 2 + 2) x 2 - 5) x 2 + 7 = (8 x 2 - 5) x 2 + 7 = 11 x 2 + 7 = 29 The calculator also computes the derivative: P’(x) = 9x² + 4x - 5, so P’(2) = 36 + 8 - 5 = 39

Polynomial vocabulary:

• Degree: The highest power of x (e.g., 3x² + 1 has degree 2)
• Leading coefficient: The coefficient of the highest-degree term
• Constant term: The term without x (a₀)
• Root/Zero: A value of x where P(x) = 0

When to use this calculator: Use it for homework checks, engineering calculations, or any time you need to quickly evaluate a polynomial expression at a specific value. It also shows the derivative value, which is useful in calculus and optimization problems.

Tips:

• Remember to include zero coefficients for missing terms
• The derivative tells you the slope (rate of change) of the polynomial at the given x value
• For quadratic polynomials (degree 2), consider using a dedicated quadratic equation solver to find the roots