## Multiplication Not Distributive (MND)

In precalculus/algebra we become familiar with the distributive laws that address interactions between multiplication and addition/subtraction. Specifically, these laws say

We use these laws all the time, both in expanding

and in factoring

We even use it (although we don't often think about it this way and usually don't include the middle step below) when combining like terms, as in

The problem is when students misinterpret these laws, thinking they also say something about interactions between more than one multiplication; that is, they invent" for themselves a law that looks something like:

This clearly is false, as most would see if these were all numbers -- few (though I cannot go so far as to say no one) would assert, say, that

But when the objects involved are expressions involving variables, the error is frequently made, such as in this case:

or

Top Algebra Errors Made by Calculus Students (full document)