Polynomial Operations Calculator
Perform addition, subtraction, multiplication, and long division on polynomials. Features symbolic computation, remainder visualization, and step-by-step logic for algebraic homework.
About
Algebraic precision is non-negotiable when dealing with high-degree functions in calculus or control theory. A single sign error in the third term of a polynomial multiplication can propagate, invalidating an entire stability analysis or integral calculation. This tool automates the arithmetic of polynomial rings, specifically focusing on the interactions between coefficients and exponents up to the 20th degree.
Students and engineers frequently struggle with the bookkeeping required for polynomial long division or the tediousness of expanding large binomials. Unlike standard calculators that output a floating-point approximation, this engine preserves the symbolic integer or fraction structure of coefficients. It visualizes the quotient and remainder separately, adhering to the Euclidean division algorithm. This distinction is critical when analyzing asymptotic behavior or performing partial fraction decomposition.
Formulas
For two polynomials A(x) and B(x), the division algorithm states:
Where the degree of the remainder R(x) is strictly less than the degree of the divisor B(x). In multiplication, the coefficient of the k-th term is given by the convolution sum:
Reference Data
| Operation | Mathematical Definition | Degree Result (deg) | Complexity Note |
|---|---|---|---|
| Addition | P+Q | max(deg(P), deg(Q)) | Combines coefficients of like powers. |
| Subtraction | P−Q | max(deg(P), deg(Q)) | Sign distribution is the primary error source. |
| Multiplication | P⋅Q | deg(P) + deg(Q) | Convolution of coefficient sequences. |
| Division (Euclidean) | P = D⋅Q + R | deg(P) − deg(D) | Requires D ≠ 0. |
| Synthetic Division | Shortcut for Linear Divisors | deg(P) − 1 | Only valid when divisor is (x − c). |
| Derivative | dP/dx | deg(P) − 1 | Linear operator on terms. |
| Integration | ∫P dx | deg(P) + 1 | Adds integration constant C. |
| Roots | P(x) = 0 | N/A | Fundamental Theorem of Algebra applies. |