Polynomial Calculator
Evaluate, add, subtract, multiply, differentiate, or synthetically divide polynomials using exact integers, decimals, or fractions.
2x³ − 5x + 7, enter 2, 0, −5, 7.Polynomial formulas
A polynomial has the form P(x)=aₙxⁿ+…+a₁x+a₀. Evaluation uses Horner’s method: repeatedly multiply the running value by x and add the next coefficient. The derivative is P′(x)=naₙxⁿ⁻¹+…+a₁. Synthetic division by x−c recursively multiplies each new row value by c and adds the next coefficient; the final value is the remainder P(c).
Checked examples
| Operation | Inputs | Result |
|---|---|---|
| Evaluate | P(x)=x²−3x+2; x=2 | P(2)=0 |
| Add | (x+2)+(x−2) | 2x |
| Subtract | (x²+1)−(x²−1) | 2 |
| Multiply | (x+1)(x−1) | x²−1 |
| Derivative | P(x)=3x³−2x+5 | P′(x)=9x²−2 |
| Synthetic division | (x³−6x²+11x−6)÷(x−2) | Quotient x²−4x+3; remainder 0 |
How to use this calculator
- Select the polynomial operation.
- Enter coefficients from the highest degree to the constant term.
- Include a zero wherever a power is missing.
- For two-polynomial operations, enter Q(x). For evaluation or synthetic division, enter x or c.
- Calculate to see exact fractions, polynomial form, degree, and verification steps.
Exact arithmetic and limits
- Integer, terminating-decimal, and fraction coefficients are reduced exactly.
- Each polynomial accepts 1 to 25 coefficients; each coefficient entry is limited to 60 characters.
- Scientific notation, mixed numbers, commas inside thousands, variables, and implicit expressions are not accepted as coefficient input.
- Multiplication may increase the degree and coefficient size substantially.
- Synthetic division here uses a linear divisor of the form
x−c.
Frequently asked questions
Why must I enter zero coefficients?
Position determines the power of x. Omitting a zero would shift every following coefficient to the wrong degree.
How do I divide by x+3?
Write the divisor as x−c. Since x+3=x−(−3), enter c=−3.
What does a zero synthetic-division remainder mean?
It means P(c)=0, so c is a root and x−c is an exact factor of the polynomial.
Does the calculator find every root?
No. It evaluates and operates on polynomials and divides by a chosen linear factor. Use the Quadratic Formula Calculator for degree-two roots.
Related calculators
Use the Quadratic Formula Calculator for quadratic roots, the System of Equations Calculator for simultaneous linear equations, the Long Division Calculator for numerical division, or the Fraction Calculator for rational arithmetic. Use the Inequality Calculator for linear, quadratic and absolute-value inequalities in interval notation.