Polynomial Calculator
Type a polynomial equation and this calculator finds its roots and its fully factored form.
Use x as the variable and ^ for powers, e.g. x^3 - 6x^2 + 11x - 6.
How to solve a polynomial by hand
- Try to spot a common factor in every term and pull it out first.
- For a cubic or higher, test small integer values (±1, ±2, ±3…) to find one root by trial, then divide it out.
- Keep factoring the remaining polynomial (often a quadratic) until every factor is linear or irreducible.
- Set each factor equal to zero to list all the roots.
Worked example
Solve x³ − 6x² + 11x − 6 = 0. Testing x = 1 gives 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor. Dividing it out leaves x² − 5x + 6, which factors into (x − 2)(x − 3). The full factored form is (x − 1)(x − 2)(x − 3), so the roots are x = 1, 2, 3.
Factor a simpler expression in the Factoring Calculator, or see every formula worked through in the maths formula list.
الأسئلة الشائعة
What is a polynomial?
An expression made of terms added or subtracted together, each a number times a variable raised to a whole-number power, like x³ − 6x² + 11x − 6.
What are the roots of a polynomial?
The values of the variable that make the polynomial equal to zero — where its graph crosses the x-axis.
How many roots does a polynomial have?
At most as many as its degree (the highest power): a degree-3 polynomial has at most 3 roots, counting repeats and complex roots.
How is factoring related to finding roots?
Each factor (x − r) of a polynomial corresponds to a root r. Factoring x² − 5x + 6 into (x − 2)(x − 3) shows its roots are 2 and 3.
