CCSS.Math.Content.HSA-APR.B.2
The standard
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
Common Core State Standards for Mathematics
What this standard means
Students need to connect three facts about polynomials: evaluating a polynomial at a number, dividing by a linear expression, and identifying factors. They should be able to find a remainder by substitution instead of long division, then use a zero result to decide whether a linear expression is a factor.
Mastery looks like moving both ways. Given a polynomial and a value, students can find the remainder and state the factor conclusion. Given a possible factor, they can test it efficiently. Common stuck points are sign errors with x minus a, mixing up a with negative a, and thinking every remainder means a factor.
Ways to teach it
- Use algebra tiles or area boxes to divide a cubic by x minus 2, then compare the remainder to direct substitution.
- Ask students to explain why p(3)=0 tells us more than just the value of a function at one point.
- Give three possible factors for one polynomial and have students test each using substitution only.
- Connect to package design by testing whether a proposed width factor gives zero leftover volume in a polynomial model.
Plan a lesson for CCSS.Math.Content.HSA-APR.B.2
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