Polynomial Calculator
Work with polynomial operations, factors, roots, and remainders in one place.
Drag & drop or upload an image or PDF
Try an example
One tool for common polynomial work
A polynomial is built from terms with nonnegative integer exponents. Its degree, leading coefficient, zeros, and end behavior all come from the same algebraic structure.
Enter the operation in plain language when the expression alone could be ambiguous. For division, name the dividend and divisor. For roots, include an equals sign or ask the calculator to solve.
How to use the polynomial calculator
Enter the problem as written
Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.
Read the working, not only the answer
Each transformation is separated and explained so you can compare it with your own method.
Ask about any step
Continue in the same solution to request another method, check a restriction, or ask why a rule applies.
Remainder theorem
When P(x) is divided by x - c, the remainder is P(c). If P(c) = 0, then x - c is a factor.
A reliable way to work through it
Line up like powers
For addition, subtraction, and long division, keep terms ordered by degree and include zero placeholders.
Use structure before expanding
Factored form and special products can save work and reveal roots immediately.
Verify with substitution
Test suspected roots with P(c) and check division by multiplying the quotient back.
Worked example
Divide x³ - 1 by x - 1
Include missing powers before division.
Match the leading term.
Repeat with the new leading term.
Finish with zero remainder.
The quotient is x² + x + 1, with remainder 0.
Common mistakes to check
Dropping missing degree terms
Use zero coefficients so polynomial division columns stay aligned.
Combining different powers
Only terms with identical variable exponents are like terms.
Assuming every root is rational
The rational root theorem gives candidates, not a guarantee that a rational root exists.
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Questions students ask
What polynomial operations can the calculator perform?
It can add, subtract, multiply, divide, expand, factor, evaluate, and solve many polynomial equations.
Can it use synthetic division?
Yes. Synthetic division is available when dividing by a linear factor in the form x - c.
Can it find complex polynomial roots?
Yes. When exact factoring is not practical, the solution may include exact complex forms or numerical approximations.
How do I ask for the degree of a polynomial?
Enter the expression and ask “find the degree and leading coefficient.”