Polynomial Division Calculator — Divide Polynomials with Quotient & Remainder

Welcome to the polynomial division calculator, where we'll learn how to divide polynomials. You might remember that when we dealt with regular numbers, there was this handy algorithm called long division. In fact, here we have a similar one, polynomial long division, and it's just as easy. But don't worry — before we see how to do long division with polynomials, we'll slowly go through the basics and begin with dividing polynomials by monomials. Nice and easy, that's our motto.

So sit back, grab a cup of tea for the journey, and let's get to it! This calculator divides one polynomial by another, giving you the quotient, the remainder, a full step-by-step long division, and a substitution check.

🔎 If you need to divide numbers instead of polynomials, head to our long division calculator!

Polynomials, binomials, monomials

Do you remember the good ol' days when mathematics dealt only with numbers? Well, there were also squares and circles, but still, we described them using numbers. And then some clever person decided to get the alphabet involved and came up with algebra.

A polynomial is a sum of monomials. Now that is a helpful definition, isn't it? And we can go one step further: a monomial is a polynomial with only one summand. So, since we've figured all of that out, let's jump to dividing polynomials using long division.

Oh, alright, we're not that evil. Let's go through it in detail. A monomial is the product of numbers and variables with non-negative integer powers. This means that, for instance:

• 2x;
• (−3) · z³ · 0.5;
• πr²;
• a²⁰²⁰;
• ½ · 0.7654321 · 1;
• k · l · (−7x) · k

are all monomials. Observe that they don't necessarily have to contain variables. The important thing here is that a monomial cannot be the sum or difference of two expressions. (The occasional minuses above come from negative numbers, such as −3, and not from the subtraction operation.) It also cannot have square roots or functions like sine or logarithm.

So, coming back to the definition, a polynomial is a sum of monomials. This means that, in particular, every monomial is a polynomial. Below are a few other examples:

• x + 2y;
• a² + 2ab + b;
• n³ − 0.7n + 3⁄8;
• 1 + 3 + x⁵ − x⁷ + 19x⁵

Lastly, a binomial is a polynomial with two summands. For instance, the first expression above is a binomial, while the others are not.

In general, a polynomial can have any number of variables. Today, however, we focus on those that have only one variable, i.e., of the form:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀

where aₙ, aₙ₋₁, …, a₁, a₀ are numbers, which we call coefficients. By convention, the variable is usually denoted by x, but if you have any letter preferences, feel free to rebel and swap it for whichever suits your needs — this calculator lets you pick from x, y, t, s, n, and z.

Dividing polynomials by monomials

The simplest kind of polynomial division is dividing a polynomial by a monomial. Since division distributes over addition, all you have to do is divide each term of the polynomial by the monomial separately. For example:

(6x³ + 9x² − 3x) ÷ (3x) = 2x² + 3x − 1

Here we divided each of 6x³, 9x², and −3x by 3x. Notice how the powers drop by one and the coefficients are simply divided. That's all there is to dividing by a monomial!

How to divide polynomials — long division

When the divisor has more than one term, we use polynomial long division, which mirrors the long division you learned for numbers. Given a dividend P(x) and a divisor D(x), the goal is to find a quotient Q(x) and a remainder R(x) such that:

P(x) = D(x) · Q(x) + R(x)

where the degree of R(x) is strictly smaller than the degree of D(x). Written as a fraction, this is the same as:

P(x) ⁄ D(x) = Q(x) + R(x) ⁄ D(x)

How to do long division with polynomials, step by step

1. Write both polynomials in descending order of powers, filling any missing power with a zero coefficient.
2. Divide the leading term of the dividend by the leading term of the divisor. This gives the next term of the quotient.
3. Multiply the whole divisor by that quotient term.
4. Subtract the product from the current dividend to get a new, smaller dividend.
5. Repeat steps 2–4 until the degree of what's left is smaller than the degree of the divisor. Whatever remains is the remainder.

Example: using the polynomial division calculator

Let's divide x³ − 2x² + 4 by x − 2. Enter x^3 - 2x^2 + 4 as the dividend and x - 2 as the divisor. The calculator returns a quotient of x² and a remainder of 4, i.e.

x³ − 2x² + 4 = (x − 2)·x² + 4

and, in fractional form, (x³ − 2x² + 4) ⁄ (x − 2) = x² + 4 ⁄ (x − 2). The step-by-step panel shows each divide–multiply–subtract cycle, and the verification confirms that the divisor times the quotient plus the remainder reproduces the original dividend.

How to use the calculator

• Type each polynomial using ordinary notation, e.g. 2x^3 + 3x^2 - x + 5. Use ^ for powers; the multiplication sign is optional.
• Prefer coefficients? You can instead enter a space- or comma-separated list from the highest power to the lowest, e.g. 2 3 -1 5.
• Fractional coefficients such as 1/2 or decimals such as 0.5 are fully supported, and results are given as exact fractions whenever possible.
• Pick the variable letter that matches your problem — the answer is displayed using the same letter.

🙋 You may have heard about the quadratic equation, one of the most popular polynomials. Check our quadratic formula calculator to learn more about it — and once you have your quotient and remainder, factoring the result is often the natural next step.