What is the Remainder Theorem?
The Remainder Theorem provides a quick way to find the remainder when a polynomial is divided by a linear expression, without performing long division.
The Remainder Theorem
Definition
If a polynomial f(x) is divided by (x - c), then the remainder is f(c).
In other words, to find the remainder of dividing f(x) by (x - c), simply evaluate the polynomial at x = c. The result is the remainder. This works because polynomial division gives f(x) = (x - c) · q(x) + r, where q(x) is the quotient and r is the remainder. Substituting x = c gives f(c) = r.
Example 1: Basic Application
Problem:Find the remainder when f(x) = x³ - 2x² + 4x - 7 is divided by (x - 3).
Step 1: Identify c = 3 (from x - 3 = 0)
Step 2: Evaluate f(3)
f(3) = (3)³ - 2(3)² + 4(3) - 7
f(3) = 27 - 18 + 12 - 7
f(3) = 14
Answer: The remainder is 14
Example 2: Verifying a Factor
Problem:Is (x - 2) a factor of f(x) = x³ - 3x² + 4?
Step 1: Evaluate f(2)
f(2) = (2)³ - 3(2)² + 4
f(2) = 8 - 12 + 4
f(2) = 0
Answer: Since f(2) = 0, (x - 2) is a factor of f(x).
This is also known as the Factor Theorem: if f(c) = 0, then (x - c) is a factor.
Example 3: Negative Value
Problem:Find the remainder when f(x) = 2x² + 5x - 3 is divided by (x + 1).
Step 1: Rewrite (x + 1) as (x - (-1)), so c = -1
Step 2: Evaluate f(-1)
f(-1) = 2(-1)² + 5(-1) - 3
f(-1) = 2(1) - 5 - 3
f(-1) = 2 - 5 - 3 = -6
Answer: The remainder is -6
Key Takeaways
- The Remainder Theorem states: remainder = f(c) when dividing by (x - c)
- If f(c) = 0, then (x - c) is a factor of f(x) (Factor Theorem)
- For (x + a), use c = -a
- This method is much faster than polynomial long division for finding remainders
Practice with the Calculator
Use our free scientific calculator to evaluate polynomials and verify the Remainder Theorem with your own examples.
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