Chapter 23: Zero-Product Property: Situation 17 From the MACMTL–CPTM Situations Project

A student in an Algebra 1 class wrote the following solution to a homework problem:

A different student commented that 6 was a solution to the equation since 6² – 4(6) – 5 = 7, but that 12 was not.

In this Prompt, the student uses an overgeneralization of the zero-product property. The zero-product property states that if ab = 0, then a = 0 or b = 0. The only real number, n , for which the property “If ab = n , then a = n or b = n ” holds for all real values a and b , is 0. In this Prompt, the student extends the zero-product property to values of n other than 0. The inaccuracy of this extension is dealt with in Focus 1. Factors are a recurring theme in the study of quadratic polynomials, so integral factors are addressed in Focus 2 and Focus 3. These Foci consider some special cases in which the student’s proposed property and solution method do apply. Focus 4 considers this Situation from an abstract-algebra standpoint, using the fact that the set of polynomials forms a mathematical entity known as an integral domain.