What is a polynomial?
A polynomial is an algebraic expression made from constants and variables raised only to non-negative integer powers. In 3x² − 2x + 5, the values 3, −2, and 5 are coefficients; 3x², −2x, and 5 are terms; the degree is 2; 3 is the leading coefficient; and 5 is the constant term.
This calculator works with univariate polynomials in x. It expands parentheses and combines like terms before applying the selected operation, so inputs such as 2(x + 1) - (x - 3) are accepted. Expressions containing variables other than x, square roots of x, or negative powers are not polynomials and produce a clear input error.
How to use the polynomial calculator
- Enter the first polynomial. Use
^for powers, as inx^3. - For addition, subtraction, multiplication, or division, enter the second polynomial. Addition, subtraction, and multiplication also let you add more expressions.
- Select the operation and choose Calculate, or press Enter from an input.
- Read the simplified result, polynomial properties, formula, and generated calculation steps.
- Use Copy, Print, or Share to keep the solution. Recent calculations remain only in the current browser tab.
Supported polynomial operations
| Operation | Required input | Calculator output |
|---|---|---|
| Addition | Two to six polynomials | Like terms combined into one polynomial |
| Subtraction | Two to six polynomials | Each later polynomial subtracted from the first |
| Multiplication | Two to six polynomials | Expanded product in descending powers |
| Division | Dividend and nonzero divisor | Quotient, remainder, and long-division steps |
| Expand and simplify | One expression | Parentheses expanded and like terms combined |
| Factor | One polynomial | Greatest common factor and rational linear factors |
| Find zeros | One nonconstant polynomial | Exact rational/simple quadratic roots or numerical approximations |
For a broader set of expression tools, visit the algebra calculators category. The separate simplify expression calculator is useful when the expression is not limited to a polynomial in x.
Polynomial formulas used by the calculator
A polynomial in standard form is:
P(x) = anxn + an−1xn−1 + ··· + a1x + a0
For addition or subtraction, coefficients with the same exponent are combined:
ck = ak ± bk
For multiplication, every term in the first polynomial is multiplied by every term in the second:
ck = Σ aibk−i
Polynomial division follows:
P(x) = D(x)Q(x) + R(x), where the degree of R is smaller than the degree of D.
Unlike ordinary decimal arithmetic, the calculator stores entered integers, fractions, terminating decimals, and scientific notation as exact rational values. For example, 0.1x + 0.2x is displayed as 0.3x rather than a floating-point artifact.
Worked polynomial examples
Example 1: Adding polynomials
Add 2x² + 3x − 4 and x² − 5x + 7.
- Combine the x² terms: 2x² + x² = 3x².
- Combine the x terms: 3x − 5x = −2x.
- Combine the constants: −4 + 7 = 3.
Result: 3x² − 2x + 3
Example 2: Multiplying binomials
Multiply (x + 2)(x − 3).
- Distribute x: x(x − 3) = x² − 3x.
- Distribute 2: 2(x − 3) = 2x − 6.
- Combine the x terms: −3x + 2x = −x.
Result: x² − x − 6
Example 3: Polynomial division
Divide x³ − 1 by x − 1.
- x³ ÷ x = x². Subtract x³ − x² to leave x² − 1.
- x² ÷ x = x. Subtract x² − x to leave x − 1.
- x ÷ x = 1. Subtract x − 1 to leave 0.
Quotient: x² + x + 1; remainder: 0. Verification: (x − 1)(x² + x + 1) = x³ − 1.
Accepted input syntax and number formats
- Powers: Write
x^2,x^3, and so on. Exponents must be whole numbers from 0 to 50. - Implicit multiplication: Inputs such as
2x,3(x + 1), and(x + 2)(x − 2)are valid. - Fractions: Use
1/2x^2,x/3, or(2x + 1)/4. A polynomial may only be divided by a nonzero constant inside an input. - Decimals and scientific notation: Values such as
0.25xand1.5e-3x^2are accepted and converted to exact fractions internally. - Negative coefficients: Inputs such as
-x^3 + 2x - 7are supported. - Variable: Use x or X only. Multivariable expressions such as
xy + y^2require a different symbolic algebra tool.
Polynomial factorization here is over rational numbers. The calculator extracts a coefficient greatest common factor, powers of x, and rational linear factors. A factor such as x² + 1 may remain because it is irreducible over the rational numbers. For integer divisors of a number rather than algebraic factors, use the integer factor calculator.
Polynomial vs Polynomial Equation
A polynomial such as x^2 + 3x + 2 is an algebraic expression. A polynomial equation is formed when a polynomial is set equal to another expression, commonly zero, such as x^2 + 3x + 2 = 0.
This calculator performs polynomial operations and can also find the zeros of a polynomial by solving P(x) = 0. For a dedicated second-degree equation workflow, use the Quadratic Equation Solver.
Common polynomial mistakes
- Combining unlike terms: x² and x have different exponents and cannot be added as 2x³.
- Losing a subtraction sign: In P(x) − Q(x), every term of Q(x) changes sign.
- Distributing incompletely: In multiplication, each term of one polynomial must multiply every term of the other.
- Using invalid exponents: x⁻¹ and √x are not polynomial terms because their exponents are not non-negative integers.
- Leaving out zero placeholders in division: When dividing x³ − 1, think of it as x³ + 0x² + 0x − 1 so the powers stay aligned.
- Confusing factors with zeros: A factor x − r corresponds to the zero x = r, but the factor and the zero are not written the same way.
How to verify a polynomial answer
Choose a simple value such as x = 2 and evaluate both the original expression and the simplified result. Equal values are a quick check, although testing one value alone is not a formal proof. For addition, subtraction, or multiplication, repeat the check at another value such as x = −1.
For division, verify the identity dividend = divisor × quotient + remainder. The remainder must be zero or have a lower degree than the divisor. For a reported zero r, substitute r into P(x); an exact root gives P(r) = 0, while a numerical root should produce a value very close to zero. Use the exponents calculator when checking larger powers by hand.
Polynomial calculator FAQs
It adds, subtracts, multiplies, and divides univariate polynomials in x. It can also expand and simplify an expression, factor it over rational numbers, and find exact or approximate zeros. Every completed calculation includes generated steps based on the entered values.
Yes. Coefficients may be integers, negative numbers, fractions, terminating decimals, or scientific notation. These inputs are converted to exact rational values for algebraic operations, preventing common binary floating-point display errors.
Yes. Use “Add another polynomial” for addition, sequential subtraction, or multiplication. Up to six expressions can be combined in one calculation. Division always uses exactly two inputs: the dividend and divisor.
The leading term of the current dividend is divided by the leading term of the divisor. That term is added to the quotient, the corresponding product is subtracted, and the process repeats until the remainder has a lower degree than the divisor.
Factoring rewrites a polynomial as a product, such as x² − 5x + 6 = (x − 2)(x − 3). Finding zeros reports the x-values that make it equal to zero: x = 2 and x = 3 in this example.
The parser rejects any unrecognized character instead of silently ignoring it. Check for a missing parenthesis, an incomplete exponent, division by a variable expression, another variable such as y, or a negative/non-integer exponent. The inline message identifies the input and location to fix.
Rational roots and simple quadratic roots are shown exactly when possible. Roots of an unfactored higher-degree polynomial are numerical approximations and are labeled as approximate. For formal or safety-critical work, verify numerical results independently.
How this calculator calculates results
The input is tokenized and fully parsed before calculation; unsupported text is never skipped. Polynomial coefficients are stored as reduced fractions using exact integer arithmetic. Addition combines equal powers, multiplication uses coefficient convolution, division uses algebraic long division, and rational factor candidates are verified by exact substitution. When an unfactored polynomial above degree two needs numerical roots, an iterative complex-root method is used and the output is explicitly marked approximate.
Last updated: July 21, 2026. Maintained by the Utilixea Editorial Team. This is an educational mathematics tool, not a substitute for independent verification in safety-critical work.
Ready to calculate? Return to the calculator, enter a polynomial in x, choose the required operation, and select Calculate to see the simplified answer and its steps.
