Polynomial Division Calculator

What this calculator does

So, you want to divide one polynomial by another, the way you would divide whole numbers, and get a quotient and a remainder. This tool does polynomial long division. You set the degree of each polynomial, enter their coefficients, and it returns the quotient, the remainder, and the division written out.

Dropdowns choose the degree of the top polynomial P and the bottom polynomial Q, and coefficient boxes appear for each.

How to use it

1. Choose the degree of P(x), the polynomial being divided.
2. Choose the degree of Q(x), the polynomial you are dividing by.
3. Fill in every coefficient for both, then press Calculate.

What polynomial long division is

Polynomial long division works just like the long division you learned for numbers, but with polynomials instead of digits. When one polynomial does not divide neatly into another, you still get a sensible answer: a quotient, the main result, and a remainder, the leftover piece that would not divide. It is the same idea as dividing 17 by 5 to get 3 with a remainder of 2. The tool takes the top polynomial, called the dividend, and the bottom one, the divisor, and carries out this process term by term.

Quotient and remainder

The result of the division has two parts, and the tool reports both. The quotient is the polynomial you get from dividing, the main answer. The remainder is what is left over, a polynomial of lower degree than the divisor that could not be divided further. The whole division can then be written as the dividend over the divisor equals the quotient plus the remainder over the divisor. If the remainder comes out as zero, the divisor divided the dividend exactly, which means it is a factor of it, just as a remainder of zero for numbers means one divides the other.

How the algorithm works

The method repeats a short cycle. Divide the leading term of the dividend by the leading term of the divisor to get the next piece of the quotient. Multiply the whole divisor by that piece, and subtract the result from the dividend, which cancels the leading term and leaves a smaller polynomial. Then repeat with what remains. Each pass lowers the degree of the leftover, and the process stops once the leftover has a lower degree than the divisor. Whatever is left at that point is the remainder. The tool runs this loop and collects the quotient pieces as it goes.

A worked example

Divide x squared plus 3x plus 5 by x plus 1. Dividing the leading terms, x squared over x gives x, the first quotient term. Multiply x plus 1 by x to get x squared plus x, and subtract, leaving 2x plus 5. Now 2x over x gives 2, the next quotient term. Multiply x plus 1 by 2 to get 2x plus 2, and subtract, leaving 3. So the quotient is x plus 2 and the remainder is 3, meaning the expression equals x plus 2 plus 3 over x plus 1.

Why the divisor cannot be the bigger one

The tool asks that the divisor Q have a degree no larger than the dividend P, and there is a reason. If the bottom polynomial is of higher degree than the top one, then it simply does not go into it at all, in the same way that 5 does not divide into 3 as a whole number. In that case the division cannot proceed: the quotient would be zero and the whole thing would just be the original fraction. So the tool asks for a divisor that is the same degree or smaller, where there is a genuine division to carry out.

Questions people ask

What is polynomial long division?

Dividing one polynomial by another the way you divide numbers, producing a quotient and a remainder.

What are the quotient and remainder?

The quotient is the main result of dividing; the remainder is the leftover polynomial of lower degree than the divisor that could not be divided further.

What does a remainder of zero mean?

That the divisor divides the dividend exactly, so it is a factor of it, just like a remainder of zero for numbers.

Why must the divisor's degree be no larger?

Because a higher-degree divisor does not go into the dividend at all, leaving a zero quotient and nothing to divide.

How can I check the answer?

Multiply the quotient by the divisor and add the remainder; it should rebuild the original dividend. The FOIL calculator helps with the multiplication of two-term factors.

References

On polynomial long division. Dividing one polynomial by another yields a quotient and a remainder of lower degree than the divisor.

1. Eric W. Weisstein, "Long Division," from MathWorld, a Wolfram resource, on long division of numbers and of one polynomial by another.
2. Eric W. Weisstein, "Synthetic Division," from MathWorld, a Wolfram resource, on a shortcut method for dividing polynomials.