Add, subtract, or multiply two polynomials and get the simplified result instantly.
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The polynomial calculator performs the three basic operations on two algebraic polynomials of any degree, automatically simplifying the result. When adding or subtracting, only the coefficients of like terms are added or subtracted, while the remaining terms stay as they are. When multiplying, every term in the first polynomial is multiplied by every term in the second, then like terms in the final result are combined. Simply write each polynomial as a text expression like 3x^2+2x-5, and the calculator automatically parses and breaks it down into its coefficients and exponents before performing the requested operation. This parsing step is what allows the calculator to accept polynomials in a natural, familiar written format rather than requiring coefficients to be entered separately for every possible power of x, which would be considerably less convenient for anything beyond the simplest expressions. Subtraction is handled internally as adding the negative of the second polynomial, which keeps the same term-by-term matching logic used for addition without requiring an entirely separate set of rules.
Polynomial arithmetic follows a consistent underlying principle across all three basic operations: only terms with the exact same power of x can be combined directly, a rule known as combining 'like terms'. A term with x² can never be combined with a term with x¹ any more than three apples can be combined with three oranges into a single count of 'six' — they are fundamentally different kinds of quantities represented by the same variable.
Addition and subtraction of polynomials are the most direct application of this rule: each term in one polynomial is matched with the corresponding term of the same power in the other, and only their coefficients are added or subtracted, while unmatched terms (a power of x present in one polynomial but not the other) simply carry through to the result unchanged.
Multiplication requires a different, more involved process because every term in the first polynomial must be multiplied against every term in the second — a technique often remembered by the acronym FOIL (First, Outer, Inner, Last) for the simplest case of multiplying two two-term polynomials, though the same distributive principle extends to polynomials of any length. Multiplying x by x uses the exponent rule that adding powers of the same base multiplies the terms, so x² times x³ becomes x⁵, not x⁶ — a frequent point of confusion worth double-checking.
After every term has been multiplied against every other term, the resulting long expression typically contains multiple terms sharing the same power of x, which must then be combined exactly as in the addition and subtraction case, collapsing what might initially look like a dozen separate terms down to a much shorter, fully simplified final polynomial.
Polynomial arithmetic is not merely an abstract algebra exercise — it is the mathematical foundation beneath modeling any relationship where a quantity depends on another raised to multiple powers: projectile motion in physics (position as a polynomial function of time), revenue and cost modeling in business (where price and quantity relationships often involve squared or cubed terms), and curve-fitting in statistics and engineering, where a dataset is approximated by a polynomial of an appropriate degree to capture its overall trend.
Write it naturally like 3x^2+2x-5, using ^ for exponents and no spaces required between terms.
Terms with the same power of x are combined into a single term by adding or subtracting their coefficients; terms with different powers stay separate.
Yes, the calculator handles polynomials of any degree, and multiplication works correctly regardless of how many terms each one has.