Polynomial Identity Expansion Calculator
Expand algebraic expressions using 10 standard polynomial identities. Features pattern recognition and step-by-step formula application.
About
Algebraic expansion often involves repetitive application of standard identities. Manual calculation introduces risks of sign errors or exponent mismanagement. This tool automates the expansion of binomials and trinomials by pattern matching against a library of standard polynomial identities. It parses the input structure to detect forms such as the square of a sum or difference of cubes. The engine isolates terms, applies the specific geometric formula, and recombines them into the final polynomial. This utility serves students verifying homework and engineers needing quick symbolic expansion without launching heavy computer algebra systems.
Formulas
The calculator uses pattern matching to identify the structure of the input expression. For a standard binomial square, the logic follows:
When coefficients are present, such as in (2x), the exponent applies to both the scalar and the variable:
Reference Data
| Identity Name | Standard Form | Expansion Result |
|---|---|---|
| Square of Sum | (a + b)2 | a2 + 2ab + b2 |
| Square of Difference | (a − b)2 | a2 − 2ab + b2 |
| Difference of Squares | (a + b)(a − b) | a2 − b2 |
| Cube of Sum | (a + b)3 | a3 + 3a2b + 3ab2 + b3 |
| Cube of Difference | (a − b)3 | a3 − 3a2b + 3ab2 − b3 |
| Sum of Cubes | (a + b)(a2 − ab + b2) | a3 + b3 |
| Difference of Cubes | (a − b)(a2 + ab + b2) | a3 − b3 |
| Square of Trinomial | (a + b + c)2 | a2 + b2 + c2 + 2(ab + bc + ca) |