Polynomial Calculator
Enter one or two polynomials and choose an operation: add, subtract, multiply, long division with quotient and remainder, evaluation at a value, or roots. Coefficients stay exact fractions, the working is written out the way a textbook shows it, and the roots are split into exact rational roots and numerical approximations. Free, no sign-up.
Write it as you would on paper, for example 2x^3 − 3x^2 + 4x − 5 or (x + 2)(x − 1). Powers use ^ or superscripts.
Result
2x² + x + 6
Quotient
7
Remainder
Working
Divide P by Q with long division.
(2x³ − 3x² + 4x − 5) ÷ (x − 2)
Round 1: divide the leading terms, 2x³ ÷ x = 2x², multiply it by Q, and subtract.
2x² × (x − 2) = 2x³ − 4x² Left over: x² + 4x − 5
Round 2: divide the leading terms, x² ÷ x = x, multiply it by Q, and subtract.
x × (x − 2) = x² − 2x Left over: 6x − 5
Round 3: divide the leading terms, 6x ÷ x = 6, multiply it by Q, and subtract.
6 × (x − 2) = 6x − 12 Left over: 7
The degree of what is left is lower than the degree of Q, so the division stops.
Quotient: 2x² + x + 6 Remainder: 7
Check: (x − 2) × (2x² + x + 6) + 7 = 2x³ − 3x² + 4x − 5
Synthetic division (Horner's scheme) by (x − 2): bring down the first coefficient, multiply by 2, add to the next coefficient, and repeat. The last total is the remainder.
| Degree | 3 | 2 | 1 | 0 |
|---|---|---|---|---|
| Coefficients | 2 | −3 | 4 | −5 |
| Multiply by the value | 4 | 2 | 12 | |
| Add down | 2 | 1 | 6 | 7 |
How to Use the Polynomial Calculator
- 1. Type the polynomial. Write it normally: 2x^3 − 3x^2 + 4x − 5. Brackets work too, so (x + 2)(x − 1) is expanded for you. Use one letter throughout.
- 2. Choose the operation. Add, subtract, multiply or divide P by a second polynomial Q, evaluate P at a value, or find the roots of P.
- 3. Read the result. It is shown in standard form, highest power first, with fractions kept exact. The degree, leading coefficient and constant term come with it.
- 4. Follow the working. Like terms are lined up column by column, partial products are listed, and long division shows every round.
Adding, Subtracting and Multiplying
To add polynomials, combine like terms — terms with the same power of the variable. To subtract, first change the sign of every term in the second polynomial, then add. For (3x² − 2x + 5) + (x³ + 2x − 1) the x terms cancel and the result is x³ + 3x² + 4.
To multiply, multiply every term of the first polynomial by the whole second polynomial, then add the partial products. For (x + 2)(x² − 3x + 1):
x × (x² − 3x + 1) = x³ − 3x² + x 2 × (x² − 3x + 1) = 2x² − 6x + 2 sum: x³ − x² − 5x + 2
The degree of a product is the sum of the degrees (1 + 2 = 3 here); the degree of a sum is at most the larger of the two.
Polynomial Long Division
Long division works like the arithmetic kind. Divide the leading term of what is left by the leading term of the divisor, write that as the next term of the quotient, multiply it back by the divisor, subtract, and repeat until what is left has a lower degree than the divisor. That final piece is the remainder. For (2x³ − 3x² + 4x − 5) ÷ (x − 2):
| Round | Divide leading terms | Multiply and subtract | Left over |
|---|---|---|---|
| 1 | 2x³ ÷ x = 2x² | 2x²(x − 2) = 2x³ − 4x² | x² + 4x − 5 |
| 2 | x² ÷ x = x | x(x − 2) = x² − 2x | 6x − 5 |
| 3 | 6x ÷ x = 6 | 6(x − 2) = 6x − 12 | 7 |
The quotient is 2x² + x + 6 and the remainder is 7, so 2x³ − 3x² + 4x − 5 = (x − 2)(2x² + x + 6) + 7. When the divisor has the form x − c, synthetic division does the same job with just the coefficients, and the calculator prints that table too.
Evaluating and the Remainder Theorem
Evaluating P at x = 2 gives 2(8) − 3(4) + 4(2) − 5 = 7 — the same number as the remainder above. That is the remainder theorem: dividing P by (x − c) leaves a remainder equal to P(c). It follows that (x − c) is a factor of P exactly when P(c) = 0.
Finding Roots
A root of P is a value of x that makes P(x) = 0. The calculator first looks for rational roots exactly. By the rational root theorem, if the coefficients are whole numbers then any rational root is a factor of the constant term divided by a factor of the leading coefficient. Each root found is divided out, with its multiplicity. Whatever is left is solved numerically. For x³ − 6x² + 11x − 6 the roots are exactly 1, 2 and 3; for x³ − 2 there is no rational root, and the numerical roots are x ≈ 1.2599 and x ≈ −0.63 ± 1.0911i.
How many roots?
A polynomial of degree n has exactly n roots counted with multiplicity once complex numbers are allowed (the fundamental theorem of algebra). Roots marked ≈ are approximations, so use them as a check rather than an exact answer.
Common Mistakes
- Not changing every sign when subtracting. −(x² − 3x + 1) is −x² + 3x − 1, not −x² − 3x + 1.
- Combining unlike terms. 3x² and 3x are different powers and stay separate.
- Skipping a missing power in division. Write x³ + 1 as x³ + 0x² + 0x + 1 on paper so the columns line up; the calculator handles this for you.
- Stopping division early. Keep going until the left-over has a lower degree than the divisor.
- Treating a numerical root as exact. 1.2599 is an approximation of ∛2.
Frequently Asked Questions
- How do you divide polynomials?
- Use long division: divide the leading term of the dividend by the leading term of the divisor, multiply the result by the whole divisor, subtract, and repeat on what is left. Stop when the remainder has a lower degree than the divisor. The calculator shows each round.
- What is synthetic division?
- A shortcut for dividing by a linear factor x − c, using only the coefficients. You bring down the first coefficient, multiply by c, add to the next coefficient, and repeat. The last number is the remainder, and the others are the quotient's coefficients.
- What does the remainder theorem say?
- When a polynomial P(x) is divided by x − c, the remainder is P(c). So evaluating the polynomial at c is a quick way to find the remainder, and P(c) = 0 means x − c is a factor.
- How are the roots of a polynomial found?
- Rational roots come first, using the rational root theorem, and each is divided out. For what remains, the calculator solves a degree-1 or degree-2 factor directly and uses an iterative numerical method (Durand–Kerner) for higher degrees. Exact roots are shown as fractions and numerical ones with a ≈ sign.
- What is the degree of a polynomial?
- The highest power of the variable with a non-zero coefficient. 3x⁴ − x + 2 has degree 4. A non-zero constant has degree 0, and the zero polynomial has no degree.
- Can I use fractions or decimals as coefficients?
- Yes. Coefficients such as 1/2 or 0.25 are read exactly and the result is written as an exact fraction, or as a decimal when you typed decimals.
- How can I factor a polynomial?
- Find a root c and divide by (x − c), or look for a common factor, difference of squares or trinomial pattern. The Factoring Calculator works through those methods with steps.