Polynomial Calculator
A polynomial is a sum of terms such as 2x² − 3x + 5. Adding and subtracting combine like terms, and multiplying distributes every term of one polynomial across the other. Enter each polynomial as a list of coefficients, from the highest power down to the constant, and this calculator returns the simplified result, its degree and its value at any x.
How to use the polynomial calculator
- Write each polynomial as coefficients, highest power first. Include zeros for missing terms: x³ + 1 is 1, 0, 0, 1.
- Choose add, subtract or multiply.
- Optionally enter an x value to evaluate the result there, then read the answer.
Formula
Add/subtract: combine coefficients of the same power. Multiply: the coefficient of xᵏ in the product is the sum of aᵢ × bⱼ over all pairs with matching powers (i + j = k). The degree of a product is the sum of the degrees; of a sum, at most the larger degree.
Worked examples
Multiplying
(2x² − 3x + 5)(x + 4) = 2x³ + 8x² − 3x² − 12x + 5x + 20 = 2x³ + 5x² − 7x + 20. At x = 1 the result is 20.
A difference of squares
(x + 2)(x − 2) = x² − 4: the middle terms cancel.
Working with polynomials
Terminology
| Term | Meaning | Example in 2x³ − 5x + 7 |
|---|---|---|
| Coefficient | The number multiplying a power | 2, −5 and 7 |
| Degree | The highest power | 3 |
| Leading term | The term with the highest power | 2x³ |
| Constant term | The term with no x | 7 |
Special products to recognise
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
Evaluating efficiently
The calculator evaluates using Horner’s method: 2x³ + 5x² − 7x + 20 = ((2x + 5)x − 7)x + 20. It needs one multiplication per coefficient and is far less error-prone by hand.
Why polynomials matter
They model areas, volumes, motion and curves, and they approximate more complicated functions. Multiplying polynomials is also the reverse of factoring, so practise both together.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Add 3x² + 2x − 1 and x² − 4x + 6.
Show answer
Sum: 4x² − 2x + 5
Common mistakes to avoid
- Leaving out zero coefficients. x² + 1 must be entered as 1, 0, 1.
- Adding unlike terms, such as x² + x, which cannot combine.
- Forgetting to distribute to every term when multiplying.
Frequently asked questions
What is a polynomial?
An expression made of terms with whole-number powers of a variable, added or subtracted, such as 3x⁴ − x + 7.
How do I find the degree of a polynomial?
It is the highest power of the variable with a non-zero coefficient.
How do I enter a missing term?
Use 0 as its coefficient, so x³ + 5 is 1, 0, 0, 5.
How do I factor a polynomial?
See how to solve quadratic equations for factoring quadratics.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.