Polynomial Calculator (Division and Synthetic Division)

Division, synthetic division, roots and factoring, with exact coefficients throughout.

A polynomial P(x) = aₙxⁿ + … + a₁x + a₀ has degree n when aₙ ≠ 0. This tool stores every coefficient as an exact fraction (integers of any size), so 0.1x + 0.2 becomes (1/10)x + 1/5 and you never see 0.30000000000000004. Dividing A by B, with B ≠ 0, is Euclidean division A = B·Q + R, where the remainder R is zero or has a smaller degree than B. In long division you divide the leading term of the running remainder by the leading term of B, multiply the result by B, subtract and repeat. Synthetic division is the shortcut for divisors of the form x − a: the remainder is P(a) (remainder theorem) and, if it is zero, x − a is a factor (factor theorem). For a divisor ax + b, divide by x + b/a and then divide the quotient by a. For roots, the tool follows the classroom recipe and goes beyond it. It first pulls out factors of x. Then it uses the rational root theorem: if p/q is a root of a polynomial with integer coefficients, p divides the constant term and q divides the leading coefficient. Each root found is removed by synthetic division, which reveals its multiplicity. When degree 2 is left, the exact quadratic formula applies, giving irrational roots like (−1 + √5)/2 and complex ones like −1/2 + (√3/2)i. When degree 4 or more is left with no rational root, Kronecker's method looks for integer factors, as in x⁴ + 4 = (x² + 2x + 2)(x² − 2x + 2). Irreducible factors of degree 3 or more get their roots from the Durand-Kerner method in 64-bit floating point. The fundamental theorem of algebra guarantees n complex roots counted with multiplicity, and Vieta's formulas tie the sum and product of the roots to the coefficients. Limits: degree up to 60 and coefficients of up to 400 digits.

Type the polynomials the way you would write them on paper.

1. On the Operations tab, enter A(x) and B(x) (for example 2x^3 - 3x + 1, x³ or (x+1)(x-2)) and choose add, subtract, multiply, divide or GCD. Division is drawn as long division.
2. On the Synthetic division tab, enter P(x) and a degree 1 divisor such as x - 2. The table shows quotient, remainder and the value P(a) from the remainder theorem.
3. On the Roots and factoring tab, see exact and decimal roots, multiplicities, the factorization over ℚ and ℂ, the rational candidate test, Vieta's formulas and the graph. Copy the result or the link.

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Sources and references for this tool

These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.

• 5.5 Zeros of Polynomial Functions (Algebra and Trigonometry 2e)OpenStax, Remainder and factor theorems, rational zero theorem, Descartes rule of signs and the fundamental theorem of algebra.
• 5.4 Dividing Polynomials (Algebra and Trigonometry 2e)OpenStax, Polynomial long division and synthetic division.

Frequently asked questions

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