Multiplying Polynomials using Area Models - Problem 3

This is a tricky multiplication problem because my second polynomial, there‘s a binomial here doesn’t have an a to the first to the first term. Like here I have a² a to the first constant. Here I have a² no a to the first and then the constant. That makes it a little tricky for some students.

I’m going to go ahead and set up my area model with 3 boxes on the one side to represent my trinomial and 2 boxes on the other side to represent my binomial and then go through and fill them in the insides and then combine like terms to find my answer.

Again what I’m really doing is just a whole bunch of distributing. This area model helps me keep track of this times both of those, this times both of those, and this times both of those. There’s like all that multiplication that goes on, and this table or this area model just helps you organize that in your brain.

So here we go, a² times a² is a to the fourth. 2a times a² is 2a to the third, 3 times a² not so bad. I’m going to go kind of quickly through this because I think you guys at this point in your homework are getting the hang of this process.

The last thing I want to show you is to look for patterns and sometimes students look for patterns diagonally. In this case they don’t show up diagonally because I didn’t have that a to the first term. So sometimes you can look diagonally for like terms, it doesn’t always work.

So what I’m going to do is write this in standard form meaning the exponents go in order from largest to smallest. My largest exponent is 4, so I start a to forth, that guy is done. Now I’m looking for my a to the third which is just this guy right here. I’m purposefully crossing these out to help me keep track in my head of what I’m putting into my final answer already. 3a² take a way 4a² is -1a², you don’t have to write the 1 take away 8a, take away 12 good.

All of those things showed up in my answer, so I know that’s the final answer. I also know it’s in standard form because my exponents go in decreasing order starting with the largest and ending with my constant term.

So this area model or table is really just a way of organizing all of this distributing stuff. You don’t have to do this. This is just one strategy. If distributing works better for you, go with that, but if the table works better, then just go with that like do what works for you. That’s one thing I like about Math is that there is often different ways to approach the same problem, and if you can find a strategy that works for you, you can really just be successful, so good luck with these, be careful with the negatives, and make sure every that shows up in your table gets accounted for in your final answer.