Lesson 7: Division on Polynomials

What exponent rules can be used to divide a polynomial by a monomial?

A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x)+r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Lesson Description This lesson bridges the gap between basic arithmetic and high-level algebraic manipulation. Students learn to navigate the rigorous, multi-step process of polynomial long division. Key components include: The Division Algorithm: Following the iterative cycle of Divide, Multiply, Subtract, and Bring Down. Setting Up the Dividend: Learning the importance of placeholders—inserting terms with zero coefficients (e.g., 0x^2) for missing powers of x to keep columns aligned. Handling Remainders: Understanding that if a division does not come out even, the remainder is expressed as a fraction over the original divisor. Checking the Work: Using multiplication (Divisor times Quotient + Remainder) to verify that the result equals the original dividend.
Purpose
The purpose of Section 13.7 is to build algorithmic persistence and structural awareness. Long division is one of the most detail-oriented processes in Elementary Algebra. Mastering this technique prepares students for MAT 137 (Intermediate Algebra) and Pre-Calculus, where finding roots of higher-degree functions and identifying asymptotes of rational functions are core requirements. It reinforces the idea that algebra is a consistent extension of the arithmetic rules students learned in grade school.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section requires a high degree of procedural complexity. Students must manage multiple variables, signs, and exponents simultaneously. A single error in the "subtraction" phase will invalidate the entire result. It requires "strategic thinking" in terms of organization (placeholders) and the ability to apply a multi-step routine to increasingly difficult problems.

Encourage students to compare intermediate results with a partner so they can ask questions and catch errors before completing the entire problem.

"you try" on page 314 and supplemental resources