Mississippi Math FA.A.34

MathGrades 9–12Algebra (A) Polynomials

The Standard

Justify why some polynomials are prime over the rational number system.

2025 Mississippi College- and Career-Readiness Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students decide whether a polynomial can be written as a product of lower-degree polynomials with rational coefficients. They justify a prime claim with a method that fits the degree.
What Mastery Looks Like: A student tests ±1 and ±2 as the only possible rational roots of x²−2, then explains why no rational root means this quadratic is prime. They factor x²−4 as (x−2)(x+2).
Common Misconceptions: Students may think x²−2 factors over the rationals because its roots, ±√2, are real. They may also think no rational roots proves a fourth-degree polynomial is prime.
How to Assess It: Exit ticket: Is x²−2 prime over the rationals? List and test every possible rational root, then explain why that settles the quadratic case.

Lesson moves

Give students cards labeled x²−2, x²−4, ±1, and ±2; have them match possible roots and calculate each substitution.
Ask: Is x⁴−4 prime because it has no rational roots? Factor it and explain your answer.
Run a discriminant sort with x²−2, x²−4, and x²+1; students label each prime or factorable over the rationals.
Give students a square garden with area 2 m²; have them write x²−2=0 and explain why no rational side length works.

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