Section 3.3: Remainder & Factor Theorems
This section introduces the Remainder Theorem and Factor Theorem, and how to use them to evaluate polynomials and identify factors.
Example 1: Using the Remainder Theorem
Find the remainder when \( f(x) = 2x^3 - 3x^2 + x - 5 \) is divided by \( x - 2 \).
Step 1: Substitute x = 2 into f(x): \( f(2) = 2(2)^3 -3(2)^2 + 2 -5 = 16 -12 +2 -5 = 1 \)
Step 2: Remainder = 1
Example 2: Factor Theorem
Determine if \( x - 1 \) is a factor of \( g(x) = x^3 - 4x^2 + x + 6 \).
Step 1: Substitute x = 1 into g(x): \( g(1) = 1 - 4 + 1 + 6 = 4 \)
Step 2: Since g(1) ≠ 0 → x - 1 is NOT a factor
Example 3: Find a Factor Using Factor Theorem
Find a factor of \( h(x) = x^3 - 6x^2 + 11x - 6 \).
Step 1: Test possible roots ±1, ±2, ±3, ±6
Step 2: h(1) = 1 -6 +11 -6 = 0 → x - 1 is a factor
Practice Problems
- Use the Remainder Theorem: f(x) = x^3 + 2x^2 -5x +3, divide by x -1
- Check if x + 2 is a factor of g(x) = 2x^3 + 3x^2 - 8x -12
- Find a factor of h(x) = x^3 -7x +6 using Factor Theorem
- Use Factor Theorem to test if x -3 is a factor of k(x) = x^3 - 2x^2 - 9x +18
- Evaluate remainder of f(x) = 3x^3 - x^2 + 4x -2 when divided by x +1