Polynomial Calculator
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Type one or two polynomials in x and choose an operation. For example (x² + 3x − 2) × (x − 1) = x³ + 2x² − 5x + 2, and dividing x³ − 1 by x − 1 gives x² + x + 1 with remainder 0.
Write powers with ^, as in 2x^3 – x + 5. Brackets are fine: (x + 1)^3.
| Polynomials | |
| P(x) | x² + 3x − 2 |
| Degree of P | 2 |
| Q(x) | x − 1 |
| Degree of Q | 1 |
| Result | |
| Answer | x³ + 2x² − 5x + 2 |
| Degree | 3 |
Show steps
- Multiply every term of P by every term of Q, then combine like terms.
How to use the polynomial calculator
- Choose an operation.
- Type the polynomials using x, numbers, + − × ÷, brackets and ^ for powers: 3x^2 – 5x + 2 or (x + 2)^3.
- Press Calculate. Results are in standard form, from the highest power down.
Polynomial operations
- Add or subtract: combine like terms. (3x² + 2x) + (x² − 5x + 1) = 4x² − 3x + 1.
- Multiply: multiply every term by every term. (x + 2)(x − 3) = x² − 3x + 2x − 6 = x² − x − 6.
- Divide: long division gives a quotient and a remainder, with P = Q × quotient + remainder. (x² + 5x + 6) ÷ (x + 2) = x + 3.
- Evaluate: put the number in for x. If P(x) = 2x² − 3, then P(4) = 29.
Roots of a polynomial
A polynomial of degree n has exactly n roots when you count complex roots and repeated roots (the fundamental theorem of algebra). Complex roots of a polynomial with real coefficients come in pairs a ± bi. So a cubic always has at least one real root.
| Polynomial | Roots |
|---|---|
| x² − 5x + 6 | 2, 3 |
| x² + 1 | i, −i |
| x³ − 1 | 1, −½ ± (√3⁄2)i |
| (x − 2)² | 2 (twice) |
Polynomial practice questions
Test yourself with 6 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you multiply polynomials?
Multiply each term of the first polynomial by each term of the second, then add like terms. (x + 1)(x² − x + 1) = x³ + 1.
How do you divide polynomials?
Use long division: divide the leading terms, multiply, subtract, and repeat until the remainder has a lower degree than the divisor.
How are the roots found?
The calculator uses the Durand–Kerner method, which finds every root at once by improving guesses until they stop changing. It works for real and complex roots.
What is standard form?
A polynomial written with its terms in order from the highest power of x to the lowest, such as 3x³ − 2x + 5.
Sources
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