The Remainder Theorem

The Remainder Theorem is a shortcut for finding the remainder of a polynomial division without doing the long division. If you divide a polynomial f(x) by (x - a), the remainder is just f(a). You plug in a value and read off the answer.

What the Theorem Says

When you divide any polynomial f(x) by a linear factor (x - a), you get a quotient and a remainder. That remainder is always a constant, and it equals f(a). So instead of doing the full division, you substitute a into the polynomial directly.

If f(x) is divided by (x - a), then:
Remainder = f(a)

Why It Works

The polynomial division identity says f(x) = (x - a) * q(x) + r, where q(x) is the quotient and r is the remainder. If you substitute x = a, the first term becomes zero because (a - a) = 0. That leaves f(a) = r, so the remainder is f(a).

Worked Example 1

Find the remainder when f(x) = x³ - 4x² + x + 6 is divided by (x - 2).

Calculate f(2):

f(2) = (2)³ - 4(2)² + (2) + 6
f(2) = 8 - 16 + 2 + 6
f(2) = 0

Remainder = 0

A remainder of 0 means (x - 2) is a factor. Checking: x³ - 4x² + x + 6 = (x - 2)(x² - 2x - 3).

Worked Example 2

Find the remainder when f(x) = 2x³ + 3x² - x + 5 is divided by (x + 1). Note that (x + 1) = (x - (-1)), so a = -1.

Calculate f(-1):

f(-1) = 2(-1)³ + 3(-1)² - (-1) + 5
f(-1) = 2(-1) + 3(1) + 1 + 5
f(-1) = -2 + 3 + 1 + 5
f(-1) = 7

Remainder = 7

Connection to the Factor Theorem

The Factor Theorem follows directly from the Remainder Theorem. It says (x - a) is a factor of f(x) if and only if f(a) = 0. When the remainder is zero, the divisor goes in evenly, making it a factor.

This is useful for factoring higher-degree polynomials. Test candidate values by evaluating the polynomial at those points. If the result is zero, you found a factor.

Practical Uses

Checking roots: Quickly test whether a given value is a root without full factoring.
Faster divisions: Skip long division when you only need the remainder.
Finding factors: Combined with the Factor Theorem, it helps break down complex polynomials.
Number theory: The idea mirrors modular arithmetic and appears in computer science.

When It Does Not Apply

The Remainder Theorem only works when dividing by a linear binomial (x - a). If the divisor is quadratic or higher degree, you need long division or synthetic division instead. The remainder in those cases is not simply f(a).