The remainder formula depends on what is being divided. For integer division, the division algorithm relates the dividend, divisor, quotient, and remainder. For polynomial division, the remainder theorem can find the remainder without carrying out the full division.

Integer remainder formula

Dividend = Divisor × Quotient + Remainder

For a positive divisor, the usual remainder is a whole number from zero up to one less than the divisor. For example, 17 divided by 5 gives quotient 3 and remainder 2, because 17 = 5 × 3 + 2.

Polynomial remainder theorem

If a polynomial f(x) is divided by x − a, the remainder is f(a). This works because the divisor is linear. For example, if f(x) = x² + 3x + 2 is divided by x − 1, evaluate f(1): 1 + 3 + 2 = 6. The remainder is 6.

How to calculate a remainder

  1. Identify whether the problem uses integers or polynomials.
  2. For integer division, divide to find the quotient and multiply it by the divisor.
  3. Subtract that product from the dividend to get the remainder.
  4. For polynomial division by x − a, evaluate the polynomial at a.
  5. Check the result using the original division relationship.

Important details

Do not confuse the remainder with the quotient. The quotient counts complete groups; the remainder is what is left. For polynomial division by a higher-degree polynomial, the remainder theorem in this simple form does not directly give the full remainder. Division conventions for negative integers can vary between programming languages and mathematical contexts, so confirm the convention expected by the question or software.

Frequently asked questions

Can the remainder be larger than the divisor?

For standard division with a positive integer divisor, no. A remainder must be smaller than the divisor and non-negative.

When is the remainder zero?

When the dividend is exactly divisible by the divisor.

Why use the polynomial remainder theorem?

It finds the remainder for division by x − a using one substitution, often saving time.