Remainder Theorem: Find Polynomial Remainders Fast

The remainder theorem says that when a polynomial p(x) is divided by (x - a), the remainder is simply p(a), the value of the polynomial at x = a. Instead of performing a long polynomial division, you just substitute one number and do ordinary arithmetic.

This matters because polynomial division is slow and error-prone, while substitution takes seconds. The theorem turns a tedious division problem into a quick evaluation, and it is also the foundation of the factor theorem, which is used to factorise higher-degree polynomials.

Key Concepts

• Dividend: the polynomial being divided, p(x).
• Divisor: here always a linear expression of the form (x - a).
• Quotient and remainder: the division gives p(x) = (x - a) times q(x) + R, where q(x) is the quotient and R is the remainder, a constant.
• Linear divisor: a divisor of degree 1. The theorem applies only to divisors of this form.

Formulas

• Remainder theorem: if p(x) is divided by (x - a), then remainder R = p(a). Assumes the divisor is exactly (x - a) with the coefficient of x equal to 1.
• Divisor (ax + b): rewrite it as a(x + b/a), so the remainder equals p(-b/a). The x inside the bracket must be isolated first.

Worked Example

Find the remainder when 2x cubed - 5x squared + 3x + 7 is divided by (x - 2).

1. The divisor is (x - 2), so a = 2.
2. Evaluate p(2) = 2(2 cubed) - 5(2 squared) + 3(2) + 7.
3. Compute each term: 2 times 8 = 16; 5 times 4 = 20; 3 times 2 = 6.
4. So p(2) = 16 - 20 + 6 + 7 = 9.
5. The remainder is 9, with no long division needed.

As a sanity check, a remainder must be a constant (degree 0), which 9 is.

Common Misconceptions

• For the divisor (x + 3), a equals 3. Wrong. Write it as (x - (-3)), so a = -3 and the remainder is p(-3). Sign errors here are the most common mistake in exams.
• The theorem works for any divisor, like (x squared + 1). Wrong. It only applies when the divisor has degree 1. For higher-degree divisors you need full polynomial division.
• Remainder zero and factor theorem are different topics. They are linked: a zero remainder means (x - a) is a factor, which is exactly what the factor theorem states.

Real-World Applications

• In algebra problem-solving, the theorem quickly checks whether a guessed linear factor really divides a polynomial, speeding up factorisation.
• In computer algebra, evaluating p(a) is far cheaper than symbolic division, so software uses this idea internally.
• In coding theory and signal processing, evaluating polynomials at specific points (a generalisation of this idea) is used to encode and decode data.

To recap: the remainder on dividing p(x) by (x - a) is just p(a). Watch the sign of a carefully, remember the divisor must be linear, and use a zero remainder as your signal that you have found a factor.