How to use the polynomial arithmetic calculator
- Type Polynomial A and Polynomial B using the letter x, for example 2x^3 - 5x + 1. Powers are written x^3, and coefficients can be integers, decimals or fractions such as 1/2.
- Choose an Operation: add, subtract, multiply, divide with remainder, or find the greatest common divisor.
- Read the headline result and the coefficient table. Division also shows the quotient and the remainder.
- Open Advanced options to change the value of x at which the answer is checked against the original polynomials with exact fractions.
Formula
A x B: multiply each term of A by each term of B and collect equal powers
A = Q x B + R with the degree of R less than the degree of B
greatest common divisor: repeat A, B = B, (A mod B) until the remainder is zero, then scale to leading coefficient 1
All coefficients are kept as exact fractions, so a decimal such as 0.1 is read as 1/10 and nothing is lost to rounding. The check evaluates both sides of the identity at one value of x with exact arithmetic.
A = 2x^3 - 5x + 1 and B = x^2 + 3x - 2
The product is 2x^5 + 6x^4 - 9x^3 - 14x^2 + 13x - 2, and the sum is 2x^3 + x^2 - 2x - 1. Dividing A by B gives a quotient of 2x - 6 and a remainder of 17x - 11. Check: (x^2 + 3x - 2)(2x - 6) + 17x - 11 = 2x^3 - 5x + 1.
At x = 2, A is 7 and B is 8, and the product evaluates to 56, which equals 7 x 8. With fractions, dividing (1/3)x^2 + 3x + 1/2 by (1/2)x + 1 gives a quotient of (2/3)x + 14/3 and a remainder of -25/6. The greatest common divisor of x^3 + 2x^2 - x - 2 and x^2 - 1 is x^2 - 1.
What each operation tells you
Multiplication builds a polynomial of degree equal to the sum of the degrees. Division with remainder is the polynomial version of long division: it tells you whether B divides A evenly (remainder zero) and, if not, what is left. A common divisor reveals shared factors, which is how you simplify a fraction of polynomials.
If the remainder after dividing by x - c equals A(c) (the remainder theorem), then a zero remainder means x - c is a factor. You can test candidate roots that way, or find all roots numerically with the Polynomial Root Finder.
Limits
The variable must be x and the highest power is 40. Very long coefficients are not meant for this page. For plotting a polynomial, use the Graphing Calculator. The Quadratic Equation Solver is quicker for a single quadratic.
More math tools
For significant figures, complex numbers, repeating decimals and differential equations, try the Repeating Decimal to Fraction Converter, Differential Equation Solver and Shortest Path Calculator.
Assumptions and limits
- The only variable is x, and every power is a whole number from 0 to 40.
- Coefficients are exact: integers, decimals or fractions, never rounded.
- Division needs a divisor other than the zero polynomial.
- The greatest common divisor is scaled to have a leading coefficient of 1; over the rationals that is the usual convention.
- The example polynomials are an illustration. Replace them with your own.
Frequently asked questions
Why does the answer show fractions?
Dividing polynomials with whole-number coefficients can need fractions in the quotient. They are shown exactly, so (1/2)x means half of x, and a decimal column beside the table gives the same values as decimals.
What happens if I type 3x - 3x?
Like terms are combined, so it becomes the zero polynomial, shown as 0. Terms in any order are accepted.
Can I divide by a polynomial of higher degree?
Yes. The quotient is then zero and the remainder is A itself, which is a valid answer to the long division.
How is the check different from the answer?
The answer comes from combining coefficients. The check substitutes one value of x into both sides of the identity and compares the exact results, which catches mistakes in either place.