Polynomial standard form calculator

This calculator performs both steps on a univariate expression and returns one normalized result. It accepts implicit coefficients, integer or decimal values, exact fractions, optional multiplication signs, and nonnegative integer powers. The result also reports the polynomial degree and a structured term list, making it useful for checking homework, preparing later algebraic operations, or normalizing expressions in an automated workflow.

Enter a polynomial as ordinary signed terms

Provide one polynomial containing a single variable, such as 3x^2 + 4x - 7 + 2x^3 - x^2. Spaces are optional, and an asterisk between a coefficient and the variable is accepted, so 3*x^2 and 3x^2 mean the same thing. A missing coefficient means one: x^4 is treated as 1x^4, while -x is treated as -1x. Coefficients may be integers, decimals, or fractions such as 5/8. Exponents must be nonnegative whole numbers. The parser deliberately rejects parentheses, products of polynomial expressions, negative exponents, multiple variables, and scientific notation because those forms require expansion or represent a different algebraic object. Keeping the accepted grammar explicit makes the normalized answer predictable. If the expression uses a letter other than x, that letter is preserved in the output. Constants alone are also valid polynomials and receive degree zero unless their combined value is zero.

Understand how like terms are combined

Terms are like terms when they contain the same variable raised to the same power. The calculator reads every signed term, groups coefficients by exponent, and adds those coefficients exactly. For example, 3x^2 and -x^2 contribute coefficients three and negative one to the degree-two group, producing 2x^2. Fraction and terminating-decimal coefficients are represented internally as exact rational values, so a calculation such as 0.1x + 0.2x becomes 3/10x rather than inheriting a floating-point rounding artifact. Any group whose coefficient becomes zero is removed. This cancellation can lower the degree of the polynomial, and complete cancellation produces the zero polynomial. The response uses degree -1 for that special zero result, a common computational convention because the zero polynomial has no highest nonzero power. The structured terms array lists each surviving degree and its reduced coefficient, which is helpful when another program needs the coefficients without parsing the displayed expression again.

Read and use the standard-form result

After combining coefficients, the calculator sorts surviving terms from the greatest exponent to the smallest. It prints signs between terms, suppresses a coefficient of one on variable terms, omits an exponent of one, and reduces fractional coefficients. The primary normalized_expression field is ready to copy into a worksheet or use as a canonical display value. The degree field identifies the greatest exponent that remains after cancellation, while variable records the symbol used by the input. Standard form is especially useful before comparing two polynomials, identifying leading terms, applying long division, differentiating, or checking whether an expression has the expected degree. This capability normalizes rather than expands: it does not distribute multiplication over parentheses or evaluate the polynomial at a chosen value. Through the API, each expression costs $0.002; the same deterministic core can run in the browser. Identical valid input therefore produces identical normalized output without network access, random choices, or time-dependent behavior.