The Polynomial Remainder Theorem Made Simple

Polynomial division is one of the most error-prone topics in high school and college algebra. Writing out synthetic or long division with polynomials takes time and multiple steps. Fortunately, the Polynomial Remainder Theorem offers a brilliant shortcut.

1. The Formal Statement of the Theorem

In algebraic notation: P(x) = (x − a) × Q(x) + R, where Q(x) is the quotient polynomial and R is the constant remainder. When x = a, the term (a − a) × Q(a) = 0, leaving strictly P(a) = R.

2. Step-by-Step Worked Example

Let's find the remainder when P(x) = 2x³ − 4x² + 3x − 7 is divided by (x − 3).

• Step 1: Identify the value of a. Here, the divisor is (x − 3), so a = 3.
• Step 2: Substitute x = 3 into P(x):
  P(3) = 2(3)³ − 4(3)² + 3(3) − 7
  P(3) = 2(27) − 4(9) + 9 − 7
  P(3) = 54 − 36 + 9 − 7 = 20
• Conclusion: The remainder when P(x) is divided by (x − 3) is exactly 20.

3. Connection to the Factor Theorem

A direct consequence of the Remainder Theorem is the Factor Theorem: if P(a) = 0, then the remainder is zero, which means (x − a) is an exact factor of P(x). This is widely used in root finding and polynomial factorization.