About

Polynomial arithmetic errors propagate through every downstream calculation in engineering, physics, and signal processing. A single sign error when combining like terms can invalidate an entire control-system transfer function or structural load polynomial. This calculator parses two polynomials in standard notation, performs exact addition or subtraction by matching terms with identical exponents, and returns the fully simplified result. It handles arbitrary degree, decimal coefficients, and implicit forms such as x (coefficient 1) or −x³ (coefficient −1).

The tool assumes a single variable x and real-valued coefficients. Accuracy is limited only by IEEE 754 double-precision floating point (≈15 - 17 significant digits). Pro tip: always verify the degree of your result polynomial. If the leading terms cancel (e.g., 3x⁴ − 3x⁴), the effective degree drops, which matters for stability analysis and curve fitting.

Formulas

Given two polynomials P(x) and Q(x) expressed as sums of terms:

P(x) = m∑i=0 a_ixⁱ

Q(x) = n∑j=0 b_jx^j

For addition, the result R(x) = P(x) + Q(x) is computed by combining coefficients of matching exponents:

R(x) = max(m,n)∑k=0 (a_k + b_k)x^k

For subtraction, negate all coefficients of Q before combining:

R(x) = max(m,n)∑k=0 (a_k − b_k)x^k

Where a_k is the coefficient of x^k in P, and b_k is the coefficient of x^k in Q. Missing terms have coefficient 0. The result is then sorted by descending exponent and zero-coefficient terms are removed.

Reference Data

Operation	Rule	Example	Result
Add like terms	axⁿ + bxⁿ = (a + b)xⁿ	3x² + 5x²	8x²
Subtract like terms	axⁿ − bxⁿ = (a − b)xⁿ	7x³ − 2x³	5x³
Unlike terms	Cannot combine; keep both	4x² + 3x	4x² + 3x
Constant terms	Constants are x⁰ terms	6 + 9	15
Implicit coefficient	xⁿ means 1xⁿ	x² + 3x²	4x²
Negative implicit	−xⁿ means −1xⁿ	−x + 4x	3x
Zero result term	If a + b = 0, term vanishes	5x − 5x	0
Distribute negative	P − Q means negate all terms of Q	(2x + 1) − (3x − 4)	−x + 5
High degree	Works for any integer exponent ≥ 0	2x¹⁰ + 3x¹⁰	5x¹⁰
Decimal coefficients	Real-valued coefficients allowed	1.5x² + 2.7x²	4.2x²
Ordering	Result sorted by descending exponent	3 + 2x² + x	2x² + x + 3
Commutative property	P + Q = Q + P	Order of inputs does not change sum	Same result
Non-commutative sub	P − Q ≠ Q − P (generally)	Subtraction order matters	Signs flip

Frequently Asked Questions

Enter terms like 3x², −5x, or 7. Use the caret symbol (^) for exponents: 3x^2 - 5x + 7. Implicit coefficients are supported: x^3 is read as 1x³, and -x as −1x. Decimal coefficients like 2.5x^4 work. Spaces are optional. The variable must be lowercase x.

When like terms combine to a coefficient of exactly 0, that term is removed from the result. If all terms cancel, the result is displayed as 0. This is critical in control theory where pole-zero cancellation changes system order. The step-by-step solution explicitly shows the cancellation so you can verify it.

Yes. The parser accepts terms in any order: 5 + 3x^2 - x is valid and internally sorted by descending exponent to produce 3x² − x + 5. The result is always presented in standard form (highest degree first).

The calculator supports non-negative integer exponents of any practical size. Internally, terms are stored as objects with an exponent property, so x¹⁰⁰ requires no more memory than x². Performance remains constant because the algorithm is linear in the total number of terms across both polynomials: O(m + n).

When the leading coefficients of P and Q are equal for the highest exponent, subtraction causes them to cancel. For example, (4x³ + x) − (4x³ − 2) yields x + 2, dropping from degree 3 to degree 1. This is called degree reduction and is mathematically correct.

No. This calculator is restricted to polynomials, which by definition have non-negative integer exponents only. Expressions with x⁻¹ (Laurent series) or x^1/2 (radical expressions) are not polynomials and will produce a parse error. For such expressions, a general algebraic simplifier is needed.