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
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students evaluate a polynomial at a given number and use the result to predict division by a related linear binomial. They decide whether the binomial divides evenly or leaves a remainder.
What Mastery Looks Like
- A student can correctly match x - 5 with 5 and x + 5 with -5. The student evaluates accurately, states the remainder, and justifies whether the binomial is a factor.
Common Misconceptions
- Students often use the wrong sign, matching x + 3 with 3 instead of -3. They may identify a binomial as a factor even when substitution gives a nonzero value. Substitution and synthetic division errors can also hide correct reasoning.
How to Assess It
- Exit ticket: For p(x) = x^3 - 4x + 1, find the remainder when dividing by x - 2 and decide whether x - 2 is a factor. Show substitution, not long division.
Lesson moves
Ways to Teach It
Give pairs polynomial cards and linear-divisor cards; students substitute the matching value, then sort divisors into factors and nonfactors.
Have students explain in writing why x + 3 requires evaluating p(-3), then compare their sign reasoning with a partner.
Run a factor hunt: teams test four candidate divisors for one polynomial, earning a point for each correct calculation and justification.
Use h(t) = -5t^2 + 20t for a launched ball; students find ground times and match each time to a factor.
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Printable HSA-APR.B.2 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSA-APR.B.2, with an answer key for the teacher on its own page. No account needed.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSA-APR.A.1
Polynomial multiplication and subtraction support checking factors and carrying out the division steps used in the Remainder Theorem.
- CCSS.Math.Content.HSA-SSE.A.2
Seeing polynomial structure and rewriting expressions supports recognizing factors like x minus a when using the Remainder Theorem.
- CCSS.Math.Content.HSF-IF.A.2
Understanding p(a) as a function value supports evaluating polynomials and recognizing when that value makes x - a a factor.
What This Unlocks
Mastery here sets students up for these next.
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