What Is the Remainder Theorem?
The Remainder Theorem says: When you divide P(x) by (x - a), the remainder equals P(a).
Instead of doing long division, just plug the number a into the polynomial. The result is your remainder.
How to Use the Remainder Theorem
Step 1: Identify the divisor (x - a) and find the value of a.
Step 2: Substitute x = a into the polynomial P(x).
Step 3: Calculate. The answer is the remainder.
Example 1: Find the remainder when P(x) = 2x³ - 3x² + 4x - 5 is divided by (x - 2).
- Divisor is (x - 2), so a = 2
- P(2) = 2(2)³ - 3(2)² + 4(2) - 5
- P(2) = 2(8) - 3(4) + 8 - 5
- P(2) = 16 - 12 + 8 - 5 = 7
- Remainder = 7
Example 2: Find the remainder when P(x) = x³ + 2x² - x + 6 is divided by (x + 1).
- Divisor is (x + 1) = (x - (-1)), so a = -1
- P(-1) = (-1)³ + 2(-1)² - (-1) + 6
- P(-1) = -1 + 2 + 1 + 6 = 8
- Remainder = 8
P(3) = (3)² - 5(3) + 6 = 9 - 15 + 6 = 0
Remainder = 0. This means (x - 3) is a factor of P(x)!
The Factor Theorem
The Factor Theorem is a special case of the Remainder Theorem:
If P(a) = 0, then (x - a) is a factor of P(x).
This is incredibly useful for factoring polynomials. If you can find a value a where P(a) = 0, you've found a factor.
Example: Is (x - 2) a factor of x³ - 4x² + x + 6?
- P(2) = 8 - 16 + 2 + 6 = 0
- Since P(2) = 0, yes, (x - 2) is a factor.
Quick Reference
| Divisor | Value of a | Evaluate |
|---|---|---|
| (x - 3) | a = 3 | Find P(3) |
| (x + 2) | a = -2 | Find P(-2) |
| (x - 5) | a = 5 | Find P(5) |
| (x + 4) | a = -4 | Find P(-4) |
Why Does It Work?
When P(x) is divided by (x - a), the Division Algorithm gives us:
P(x) = (x - a) · Q(x) + R
where Q(x) is the quotient and R is the remainder (a constant). If we plug in x = a:
P(a) = (a - a) · Q(a) + R = 0 · Q(a) + R = R
So P(a) always equals the remainder R.
Common Mistakes
- Wrong sign for a: For (x + 3), a = -3 (not +3). Always write the divisor as (x - a) first.
- Arithmetic errors: Be careful with negative numbers, especially when a is negative.
- Confusing remainder with quotient: P(a) gives the remainder, not the quotient. For the quotient, you still need to divide.
See also: Dividing Polynomials by Binomials or try our Polynomial Calculator.