Overview
This lecture explains how to perform polynomial long division using the box method, including step-by-step solutions for examples with and without remainders.
Introduction to the Box Method
- The box method is a visual technique for polynomial long division, arranging the divisor along the side of a box.
- The goal is to reconstruct the dividend polynomial inside the box by filling out terms step by step.
- This method works like finding missing dimensions of a rectangle given its area.
Example 1: Division Without Remainder
- Arrange the divisor (e.g., x + 2) along the box side.
- Match the highest-degree term by finding what multiplies the divisor’s lead term to equal the dividend’s lead term.
- Fill in subsequent terms by distributing, then continue matching terms until the dividend is fully accounted for.
- The quotient is formed by the terms written at each step (e.g., 3x² + 5x + 1).
Example 2: Division With Remainder
- Proceed as in Example 1, but if a leftover constant or term remains after all steps, it's the remainder.
- The remainder is written as a fraction over the divisor (e.g., -3/(2x - 3)), added to the quotient.
Example 3: Missing Terms in Dividend
- If a term (such as the x term) is missing in the dividend, include a placeholder.
- Still perform the box method steps as usual, distributing and matching degrees as needed.
- Any remainder is written over the divisor.
Example 4: Division by a Trinomial
- When dividing by a trinomial (three-term polynomial), extend the box accordingly.
- Distribute and match descending powers, accounting for all terms, including cross-terms.
- Any remaining constant not accounted for is the remainder, written over the divisor.
Key Terms & Definitions
- Box Method — A visual approach to polynomial long division using a grid or box to organize terms.
- Dividend — The polynomial being divided.
- Divisor — The polynomial you're dividing by.
- Quotient — The result of the division, excluding the remainder.
- Remainder — The term(s) left after division, written as a fraction over the divisor.
Action Items / Next Steps
- Practice the box method on several division problems, including those with and without remainders.
- Review traditional long division for polynomials as a comparison.
- Complete any assigned homework or reading on polynomial division techniques.