liminfoPolynomial Calculator
Free web tool: Polynomial Calculator
Enter coefficients from highest to lowest degree, comma-separated. E.g. "1, 0, -4" = x^2 - 4
P(x) + Q(x) =
x^2 + x -2
Coefficients: [1, 1, -2]
About Polynomial Calculator
The Polynomial Calculator lets you perform arithmetic operations on two polynomials — addition, subtraction, and multiplication. You enter each polynomial as a comma-separated list of coefficients from the highest degree to the lowest (for example, "1, 0, -4" represents x² − 4). The tool instantly parses the input, displays the human-readable symbolic form (such as "x^2 - 4"), and computes the result of the selected operation.
In addition to arithmetic operations, the calculator automatically finds the real roots of both input polynomials. For linear polynomials (degree 1), it applies direct division. For quadratic polynomials (degree 2), it uses the quadratic formula and only returns real roots when the discriminant is non-negative. For higher-degree polynomials, it employs a numerical bisection algorithm that scans the interval [−100, 100] in steps of 0.1 to bracket sign changes, then refines each root to 6 decimal places using 50 iterations of bisection.
This tool is used by mathematics students, engineers, and scientists who need to quickly verify polynomial arithmetic, check factorizations, or find approximate roots without a CAS (Computer Algebra System). It is especially handy for control systems engineers working with characteristic polynomials, signal processing engineers dealing with transfer function numerators and denominators, and algebra students practicing polynomial operations.
Key Features
- Polynomial addition: P(x) + Q(x) with instant symbolic display of result
- Polynomial subtraction: P(x) − Q(x) with coefficient-level accuracy
- Polynomial multiplication: P(x) × Q(x) using convolution of coefficient arrays
- Automatic real root finding for P(x) and Q(x) using quadratic formula and bisection
- Coefficient input as comma-separated list from highest to lowest degree
- Symbolic polynomial display (e.g. x^2 - 4) for easy visual verification
- Raw coefficient array output for use in further calculations
- 100% browser-based — no server, works offline, handles any polynomial degree
Frequently Asked Questions
How do I enter a polynomial into this calculator?
Enter the coefficients separated by commas, from the highest degree term to the constant term. For example, x³ − 2x + 5 is entered as "1, 0, -2, 5" (coefficient of x³ is 1, x² is 0, x is −2, constant is 5). Zeros for missing terms must be explicitly included so the calculator knows the degree of each term.
How does the polynomial root finder work?
For linear polynomials (degree 1: ax + b), the root is x = −b/a. For quadratic polynomials (ax² + bx + c), the quadratic formula x = (−b ± √(b²−4ac)) / (2a) is used, returning real roots only when b²−4ac ≥ 0. For higher-degree polynomials, a numerical bisection method scans from −100 to +100 in steps of 0.1, detects sign changes (which bracket a root), then converges to each root within 6 decimal places using 50 bisection iterations.
Why does the root finder sometimes miss roots?
The numerical bisection scans the range −100 to +100. Roots outside this range will not be found. Also, if two roots are very close together (within 0.1 of each other), the algorithm may merge them or miss one. Additionally, complex (non-real) roots are not detected since bisection only works for real roots. For complete root-finding including complex roots, a dedicated CAS like Wolfram Alpha or MATLAB is recommended.
What is polynomial multiplication and when is it useful?
Polynomial multiplication (also called convolution of coefficient arrays) multiplies two polynomials term-by-term and collects like terms. It is used when expanding factored forms (e.g., (x − 2)(x + 3) = x² + x − 6), computing transfer function products in control theory, finding characteristic polynomials of block diagrams, and when working with z-transforms or Laplace transforms in signal processing.
Can this calculator handle very high-degree polynomials?
Yes, there is no hard limit on polynomial degree. However, the numerical root finder only scans [−100, 100] and uses a fixed step of 0.1, so it is most reliable for polynomials with roots in this range. For very high-degree polynomials (degree > 10), the coefficient values may become very large due to multiplication, and floating-point precision limitations may affect accuracy. Always verify high-degree results independently.
How is the result displayed symbolically?
The calculator builds a human-readable string from the coefficient array. Coefficients of ±1 for non-constant terms omit the "1" (e.g., "x" not "1x"). Zero coefficients are skipped. Positive coefficients after the first term are prefixed with "+" (e.g., "+ 3x" not "3x"). The exponent is written as "x^n" for degrees above 1 (e.g., "x^3 - 2x + 5").
What is the difference between polynomial degree and the number of coefficients?
The degree of a polynomial is the highest power of x with a non-zero coefficient. A polynomial of degree n has n+1 coefficients (from x^n down to x^0). For example, x³ + 2x − 1 has degree 3 and 4 coefficients: [1, 0, 2, −1]. When entering coefficients, include a 0 for every missing intermediate term to preserve the correct degree mapping.
Can I use this for polynomial long division?
This calculator currently supports addition, subtraction, and multiplication only. Polynomial division (long division or synthetic division) is not supported. To divide P(x) by Q(x), you can find the roots of Q(x) first using this tool, then factor Q(x) manually and perform synthetic division by hand. Alternatively, use a full CAS tool that supports polynomial division directly.