Definition: The remainder theorem states that when a polynomial f(x) is divided by x – c, the remainder equals f(c), so you can find it by evaluation.
What does the theorem say?
Divide a polynomial f(x) by x – c. The remainder is exactly f(c). No long division is needed if you only want the remainder.
Why is it true?
Division gives f(x) = (x – c)q(x) + r. Substitute x = c and the first product vanishes, leaving f(c) = r.
How do you use it?
Worked example: find the remainder when f(x) = x4 – 3x2 + 6x – 10 is divided by x + 2. Here x + 2 = x – (-2), so c = -2.
f(-2) = 16 – 12 – 12 – 10 = -18.
So the remainder is -18, and x + 2 is not a factor.
What do students confuse?
The sign of c. Dividing by x + 2 means evaluating at -2, not at 2. Mathbot reminds you to set the divisor equal to zero to find the right input. Also, a remainder of zero is special: it signals a factor, which is the factor theorem.
Where to learn it
- Dividing Polynomials: Remainder and Factor Theorems – lesson
- Polynomial Functions: Zeros, Ends, and Graphs – lesson