Overview
This lecture explains the Remainder Theorem, including its definition, how to apply it, and examples using both direct substitution and synthetic division to find the remainder when dividing polynomials.
Remainder Theorem Basics
- The Remainder Theorem states that for a polynomial p(x), when divided by (x − c), the remainder is p(c).
- To find the remainder, substitute c into the polynomial and compute the result.
- The remainder is the value left after division when the quotient is not exact.
Example 1: Dividing by x − 2
- Given p(x) = x⁴ + 4x³ − x² − 16x − 4, divisor is x − 2.
- The value indicator c is 2.
- Substitute 2 into the polynomial: p(2) = 16 + 32 − 4 − 32 − 4 = 8.
- The remainder is 8.
Example 2: Dividing by x + 2
- Given p(x) = 5x² − 2x + 1, divisor is x + 2.
- The value indicator c is −2.
- Substitute −2 into the polynomial: p(−2) = 5(4) + 4 + 1 = 20 + 4 + 1 = 25.
- The remainder is 25.
Synthetic Division Method
- For p(x) = 5x² − 2x + 1 divided by x + 2, use c = −2 in synthetic division.
- Set up coefficients: 5, −2, 1.
- Compute stepwise to get final result of 25, matching the Remainder Theorem.
Example 3: Dividing by 2x − 3
- Given p(x) = 2x⁴ + 5x³ + 2x² − 7x − 15; divisor is 2x − 3.
- Set 2x − 3 = 0 → x = 3/2.
- Substitute 3/2 into the polynomial and simplify to get remainder 6.
- Synthetic division with value 3/2 and coefficients 2, 5, 2, −7, −15, also yields remainder 6.
Key Terms & Definitions
- Remainder Theorem — A method for finding the remainder when dividing a polynomial by (x − c) by evaluating p(c).
- Synthetic Division — A shortcut method for dividing a polynomial by a linear divisor.
Action Items / Next Steps
- Practice finding remainders for different polynomials using both substitution and synthetic division.
- Review any assigned readings or problems on polynomial division.