Remainder Formula

The remainder formula is used to find the value left over when one number is divided by another. It is an important concept in arithmetic, algebra, and polynomial division.

Remainder Formula

The basic division relationship is:

Dividend = Divisor × Quotient + Remainder

Therefore, the remainder can be calculated as:

Remainder = Dividend − (Divisor × Quotient)

Example of the Remainder Formula

Suppose 17 is divided by 5. The quotient is 3.

Remainder = 17 − (5 × 3)

= 17 − 15 = 2

So, the remainder is 2.

Remainder Theorem

For a polynomial f(x) divided by x − a, the remainder can be found by evaluating the polynomial at x = a.

Remainder = f(a)

Example Using the Remainder Theorem

Find the remainder when f(x) = x² + 3x + 2 is divided by x − 2.

Since x − 2 = 0, use x = 2:

f(2) = 2² + 3(2) + 2

= 4 + 6 + 2 = 12

Therefore, the remainder is 12.

Important Rule

The remainder must always be smaller than the divisor in ordinary integer division. If the division is exact, the remainder is 0.