Polynomial Calculator — Add, Multiply, Factor & Divide

Combine
Analyze
P(x)
Use ^ for powers (or x² visually). Implicit multiplication is fine: 5x, (x+1)(x-2). x^3 + 2x^2 - 5x + 3 · (x+1)(x-2) · x^4 - 16
Q(x)
Powers: x^2 or visual x². Higher: x^5, x^{10}.
Coefficients: 3x^2 = 3 · x². Negatives: -2x. Fractions: x/2.
Operations: Add / Subtract / Multiply / Divide need both P(x) and Q(x). Expand / Factor / Roots use only P(x). Evaluate also takes a value for x.
Factored input: (x-1)(x+2)(x-3) works just as well as the expanded form.
Equations: for Factor, Roots, or Expand, you can paste an equation with =: x^5 + 9x = 10x^3 auto-rearranges to x⁵ − 10x³ + 9x = 0 before solving.
P(x)

Try an example

Add Subtract Multiply Divide Factor Roots Evaluate

or pick an operation and type your own

Calculate a polynomial to see its graph with roots marked.

Template:
Long division P(x) = Q(x)·D(x) + R(x),   deg R < deg D
Factoring x³ − 6x² + 11x − 6 = (x−1)(x−2)(x−3)
Rational Root Thm root p/q ⇒ p ∣ constant,   q ∣ leading coefficient
Working with degree-2?
Quadratic Formula Calculator →
Step-by-step formula, completing the square, factoring, inequalities, and a 50-problem worksheet.
Have an inequality?
Inequality Solver →
Linear, polynomial, rational, absolute-value, and compound inequalities with sign chart and interval notation.

Frequently asked

To add polynomials, align like terms (same power of x) and add their coefficients. Example: (x³ + 2x² − 5x + 3) + (x² − 4) = x³ + 3x² − 5x − 1. To subtract, distribute the negative sign across the second polynomial first, then combine like terms. The calculator shows each step.
Polynomial long division follows the same algorithm as numerical long division. Divide the leading term of the dividend by the leading term of the divisor to get the first quotient term. Multiply the entire divisor by that term, subtract from the dividend, then repeat with the remainder. Continue until the remainder degree is less than the divisor degree.
Start by factoring out the greatest common factor (GCF). For quadratics, find two numbers that multiply to ac and add to b, or use the quadratic formula. For higher degrees, try the Rational Root Theorem, synthetic division, or special patterns like difference of squares a² − b² = (a + b)(a − b) and sum/difference of cubes.
Yes. The Fundamental Theorem of Algebra guarantees a degree-n polynomial has exactly n roots (counted with multiplicity) over the complex numbers. The calculator uses the Nerdamer algebra engine to find both real and complex roots. For example, x² + 1 = 0 returns the roots i and −i.
The Rational Root Theorem states that any rational root p/q of a polynomial with integer coefficients must have p dividing the constant term and q dividing the leading coefficient. For x³ − 6x² + 11x − 6, possible rational roots are ±1, ±2, ±3, ±6. Testing these finds roots at x = 1, 2, 3.
Yes. Click the 📷 Scan button to upload a photo or take a picture of your polynomial problem. The AI extracts the polynomial(s) from the image, auto-detects the operation (factor, multiply, divide, etc.), and runs them through the calculator. Works for textbook pages, homework photos, or whiteboard captures.
Click the Practice Worksheet button below the result panel for 1,500+ SymPy-verified polynomial problems with full answer keys. Problems are organized by 26 types and 4 difficulty levels (basic, medium, hard, scholar). Suitable for class 9, class 10, AP precalculus, and college algebra.
26 types: addition, subtraction, multiplication (incl. binomial expansion), long division, synthetic division, factoring (quadratic, special patterns, grouping, high-degree), Rational Root Theorem, polynomial from given roots, Vieta's symmetric functions, polynomial inequalities, partial fractions, and parametric problems (find k for factor, common-factor parameters, multi-divisibility, double roots, radical roots).