Polynomial Equation Calculator
Solve polynomial equations and calculate roots, factors, long division, multiplication, subtraction and function values from standard polynomial notation.
Solve polynomial equations, factor common educational polynomials, perform polynomial long division, multiply or subtract polynomials, and evaluate polynomial functions in one calculator.
x^3 - 6x^2 + 11x - 6. The equation solver supports degree 1 through 8 and returns real and complex roots. Parenthesized expressions are not parsed directly; expand them first or use the multiplication mode.Polynomial result
Use this polynomial equation calculator as an all-in-one polynomial solver for common single-variable algebra work. You can solve polynomial equations for real and complex roots, factor common educational polynomials, perform polynomial long division, multiply or subtract polynomials, and evaluate a polynomial function at a chosen value of x. Enter expanded notation such as x^3 – 6x^2 + 11x – 6 and the result appears instantly without reloading the page.
What the polynomial solver can calculate
A polynomial equation solver should do more than return one root. The calculator separates the main tasks students and teachers search for so the inputs remain clear.
| Mode | What it calculates |
|---|---|
| Solve equation | Real and complex roots for degree 1 through 8 |
| Factor polynomial | Rational/real factors found from the roots, with any unfactored residual polynomial shown |
| Long division | Quotient, remainder and division identity P(x) = D(x)Q(x) + R(x) |
| Multiply | Expanded product with like terms combined |
| Add / subtract | Coefficient-by-coefficient combination of like powers |
| Evaluate function | P(x) at a numerical x value using Horner’s method |
Solving polynomial equations calculator method
The equation mode first moves every term to one side so the problem has the form P(x) = 0. Linear and quadratic equations are solved directly. For higher-degree equations, the calculator first removes exact rational roots when the Rational Root Theorem identifies them, then numerically solves any remaining cubic through degree-eight factor. That combination improves stability on common classroom examples with repeated or rational roots while still returning non-real roots when they exist.
Searches such as polynomials equation, solving polynomial equations calculator, and polynomial equation solver all point to this same root-finding task: rewrite the equation as P(x) = 0 and find every value of x that makes the polynomial zero.
For example, x^3 – 6x^2 + 11x – 6 = 0 has the roots 1, 2 and 3 because the polynomial factors as (x – 1)(x – 2)(x – 3).
Polynomial long division solver
The long division calculator polynomials mode uses the polynomial division algorithm described in standard algebra texts. A polynomial long division solver and a dividing polynomials solver are two names for this same quotient-and-remainder calculation. For a dividend P(x) and a non-zero divisor D(x), there are unique polynomials Q(x) and R(x) such that:
P(x) = D(x)Q(x) + R(x)
with either R(x) = 0 or the degree of the remainder lower than the degree of the divisor. This is the same relationship used by a dividing polynomials solver. The calculator reports the quotient, remainder and the leading-term division steps so you can check the work rather than seeing only the final expression.
Calculator factoring polynomials
Factoring reverses multiplication. The calculator tests rational roots first and repeatedly divides out the corresponding linear factors. When the remaining polynomial is quadratic, real quadratic roots are also used to finish the real factorization. If a higher-degree residual does not have a rational factor that the calculator can identify, it leaves that residual polynomial visible instead of pretending the factorization is exact.
The Factor Theorem explains the connection: if P(r) = 0, then (x – r) is a factor of P(x). The Rational Root Theorem narrows the possible rational roots of an integer-coefficient polynomial to ratios formed from factors of the constant and leading coefficients.
Multiplying polynomials calculators and subtraction
To multiply polynomials, every term in the first polynomial is multiplied by every term in the second and coefficients with the same exponent are combined. Searches for multiplying polynomials calculators usually refer to this expansion-and-combine process. For example, (x + 2)(x – 3) expands to x^2 – x – 6.
The subtract polynomials calculator works coefficient by coefficient. Subtracting B(x) from A(x) changes the sign of each term in B(x) before the like powers are combined. The same mode can also add two polynomials.
Polynomial function calculator
A polynomial function calculator evaluates P(x) at a particular input. This implementation uses Horner’s method, which rewrites the calculation as a sequence of multiply-and-add operations and avoids separately computing every power of x. For P(x) = 2x^3 – 3x + 5 and x = 2, the value is 15.
Cubic polynomial formula
A general cubic has the form ax^3 + bx^2 + cx + d = 0 with a ≠ 0. The classical cubic polynomial formula begins by dividing by a and substituting x = t – b/(3a) to remove the squared term. This produces a depressed cubic t^3 + pt + q = 0, which can be solved with Cardano’s formula. In practice, the calculator uses exact rational-root reduction when available and a numerical complex-root method for the remaining cubic because that is more stable and readable for a general browser calculator.
For a cubic polynomial calculator, the important result is that all three roots are returned, including complex roots when the cubic has only one real root.
Polynomial input rules
- Use x as the variable.
- Use ^ for exponents, for example x^4.
- Decimals and negative coefficients are accepted.
- Missing powers are allowed; x^3 – 2 means the x^2 and x terms have coefficient 0.
- For the equation mode, either enter P(x) = 0 or enter an equation such as x^2 = 4.
- Expand parentheses before entering a single polynomial. If you want to expand two factors, use the multiplication mode.
Frequently asked questions
What is a polynomial equation calculator?
A polynomial equation calculator finds values of x that make a polynomial equation true. This page also includes related polynomial operations so you can move between roots, factors, division and arithmetic without switching tools.
Does the polynomial equation solver return complex roots?
Yes. The solver returns real and complex roots for supported equations through degree 8. Very small numerical imaginary parts are rounded to zero when they are only floating-point noise.
Can this be used as a long division calculator polynomials tool?
Yes. Enter the dividend and divisor in expanded x notation. The result gives the quotient, remainder and the identity connecting them.
What is the difference between a polynomial solver and factoring?
Solving asks for the values of x where P(x) = 0. Factoring rewrites P(x) as a product of simpler expressions. The Factor Theorem connects the two because each root r corresponds to a factor (x – r).
Does the calculator handle every symbolic polynomial expression?
No. It is intentionally limited to expanded single-variable polynomials in x. It does not parse nested parentheses, multivariable expressions, fractional exponents or symbolic parameters.
Methodology and sources
The arithmetic and long-division logic follows the standard polynomial division algorithm. Root and factor logic uses the Factor Theorem and Rational Root Theorem for exact classroom-style roots, direct formulas for degree 1 and 2, and a numerical simultaneous-root method for remaining higher-degree equations. OpenStax and LibreTexts are used for the algebra definitions and theorem descriptions; current first-page polynomial calculators were reviewed to identify the combined search intent around solving, factoring, long division, multiplication and subtraction.