Here I have a trinomial that looks pretty darn nasty; it is nasty for a lot of reasons. One reason why it’s nasty is because my leading coefficient is not 1 it’s 20.Another reason why its nasty is because all of my coefficients are kind of big numbers so that kind of gives me a hint that maybe I have a monomial greatest common factor. Here's what I mean.
I want to think are there any numbers that multiply into 20 into 80 and into 35. And that number is 5.I can factor out 5 and then I’m like un-distributing 5 times what gives me 20x? It was 4x² excuse me 20x², 4x². 5 times what gives me 80? That's 16x and then 5 times what gives me 35? That's the 7.
Okay it’s still an ugly problem but it’s still not as bad. It’s still ugly because my leading coefficient is not 1, but at least I don't have that nasty like 20, 80, 35 so I’m in the right direction. Now I know I’m looking for things that my first terms had the product of 4x, it might be 4x and x it might be 2x and 2x I’m not sure yet and my last numbers have to multiply to 7, so I’m just going to guess something like this let's see how I did.
If I Foil in here I’ll have 4x², which is good, plus 4x plus 7x, darn that’s only 11x. That's not right, shoot. So I already know this isn't correct. I want you guys to start practicing doing this in your head so that you can do the Foil process without having to write everything out.
Instead of 4x I’m going to 4x and 1x I’m going to try 2x and 2x I don't know, I’m just guessing. Now I need things that multiply with 7 like 7 and 1. Foil in your head or on paper and see how it goes. When I Foil I’ll have 4x² good. My outers are 14x plus 2 more x that's 16x yes I did it, plus 7 good.
That told me I have the right answer when I Foiled in my head I knew that the product of my first terms was 4 the product of my last terms was 7 and when I found the inners and outers and combined those two like terms using the Foil process I got 16x.
So you guys I’m not going to lie to you these are tricky problems for a lot of students they are kind of like annoying because you have to do guess and check, but there is no way around it but please believe me that they'll get better.
You'll start getting better at this the more and more you try them. You'll start being able to Foil in your head and start being able to find products more quickly than you can when you are just first starting out. Don't give up you can do it.
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M.A. in Secondary Mathematics, Stanford University B.S., Stanford University
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