Polynomial Calculator
Add, subtract, multiply polynomials and evaluate them at any point. See the step-by-step combination of like terms.
Reviewed by the ToolNestr Editorial Team — July 2026
How polynomial calculations work
A polynomial is an algebraic expression composed of terms. Each term has a coefficient (a constant multiplier) and a variable raised to a non-negative integer exponent. The general form of a polynomial in one variable is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ is the leading coefficient and n is the degree.
Adding and subtracting polynomials is performed by combining like terms — terms that have the same variable exponent. For example, (3x² + 2x - 5) + (x² - 4x + 3) = (3+1)x² + (2-4)x + (-5+3) = 4x² - 2x - 2. The calculator shows this combination step by step so you can follow the process.
Polynomial multiplication uses the distributive property: each term in the first polynomial multiplies every term in the second. For example, (x + 2)(x + 3) = x(x+3) + 2(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6. Evaluating a polynomial means substituting a specific value for x and computing the result using standard order of operations.
The formula explained
General polynomial form
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Each aᵢ is a coefficient. The highest index n is the degree. Terms with zero coefficients are omitted.
Evaluation (Horner's method)
P(x) = a₀ + x(a₁ + x(a₂ + ...))
Horner's method evaluates polynomials efficiently using nested multiplication, reducing the number of multiplications.
Worked example: Evaluate P(x) = 3x² + 2x - 5 at x = 2
Step 1: Substitute x = 2: P(2) = 3(2)² + 2(2) - 5
Step 2: Evaluate exponents: 3(4) + 2(2) - 5
Step 3: Multiply: 12 + 4 - 5
Step 4: Add/subtract: 12 + 4 = 16, 16 - 5 = 11
Therefore: P(2) = 11
Using Horner's method: P(2) = -5 + 2(2 + 2(3)) = -5 + 2(2 + 6) = -5 + 2(8) = -5 + 16 = 11 ✓
Students & Teachers
Polynomials are a core topic in algebra courses. Understanding operations with polynomials — addition, subtraction, multiplication, and evaluation — is essential for advanced math including calculus, where polynomials are the simplest functions to differentiate and integrate.
Data Scientists
Polynomial regression models relationships between variables using polynomial functions. A quadratic (degree 2) model can capture curvature in data. Higher-degree polynomials can fit complex patterns but risk overfitting.
Engineers
Polynomials model physical systems: beam deflection (cubic polynomials), projectile motion (quadratic), and electrical filter responses. Taylor series expand any smooth function as an infinite polynomial for approximation.
Economists
Cost functions, revenue curves, and utility functions are often modeled as polynomials. Quadratic cost functions (TC = aQ² + bQ + c) capture diminishing returns. Cubic polynomials model S-shaped growth curves in production theory.
How to use the polynomial calculator
Enter coefficients
Type the coefficients separated by commas, starting from the highest degree term. For 3x² + 2x - 5, enter "3, 2, -5". Include zeros for missing terms.
Set the evaluation point
Enter the value of x at which to evaluate the polynomial. The result appears instantly with the step-by-step calculation shown.
Review the details
The calculator shows the polynomial expression, degree, leading coefficient, evaluation result, and the step-by-step substitution process.
Tips for working with polynomials
Standard form matters
Always write polynomials in standard form — terms ordered from highest degree to lowest (descending powers). This makes it easy to identify the degree, leading coefficient, and like terms for combining. The calculator expects coefficients in this order.
Missing terms are zero
If a polynomial skips a degree (e.g., 3x² + 5 has no x term), enter a zero for the missing coefficient. For 3x² + 0x - 5, you would enter "3, 0, -5". The calculator correctly handles this and omits the zero term from the display.
Check your signs
Negative coefficients must include the minus sign. The expression "3, 2, -5" represents 3x² + 2x - 5. A coefficient of "0" means that term is absent. Double-check your signs when entering coefficients for subtraction and negative terms.
Use Horner's method mentally
For quick manual evaluation, Horner's method is efficient: start with the leading coefficient, multiply by x, add the next coefficient, repeat. For 3x² + 2x - 5 at x = 2: start with 3, multiply by 2 = 6, add 2 = 8, multiply by 2 = 16, subtract 5 = 11. Only 2 multiplications and 2 additions.
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Frequently asked questions
What is a polynomial?
A polynomial is an expression consisting of variables (like x) and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. For example, 3x² + 2x - 5 is a polynomial.
What are like terms?
Like terms have the same variable raised to the same power. For example, 3x² and -2x² are like terms and can be combined: 3x² - 2x² = x². But 3x² and 2x are not like terms because they have different exponents.
How do you evaluate a polynomial?
Substitute the given value of x into the polynomial and simplify. For example, evaluating 3x² + 2x - 5 at x = 2 gives 3(4) + 4 - 5 = 12 + 4 - 5 = 11.
What is the degree of a polynomial?
The degree is the highest exponent of the variable. For example, 3x² + 2x - 5 has degree 2. A constant has degree 0. A linear polynomial has degree 1, quadratic degree 2, cubic degree 3.
What operations can I perform?
You can add polynomials (combining like terms), subtract polynomials, multiply polynomials (distributing each term), and evaluate a polynomial at a specific value of x.