When multiplying polynomials using the vertical method, you're doing the same routine as if you were multiplying two numbers, like 123 by 12. If I were to do this problem, I would take first 2 times 3, six; 2 times 2, four. 2 times 1 – 2. And then when I move to this next one I'm going to skip this spot and take 1 times each one of these, so 3, 2, 1 and then I would add everything in the columns together.
Using polynomials, we have the same mentality as we did with the numbers. We have two pieces on the bottom, 3 on the top. We're going to take 3 times each one of these and write it on the first row. So 3 times 2, 6. 3 times 2x is 6x and as you can see already the difference between this one and the numbers that we did is that you're going to have a plus or minus sign in between the terms. 3 times x squared is just 3x squared.
All right now we move to the x, we're going to skip this spot and we're going to right underneath here. X times 2 is 2x. And that makes sense because when we go to add down, we want to make sure that everything – our like terms – are in one column. X times 2x, 2x squared. X times x, x cubed. So we're going to go ahead and add straight down, I get a positive 6, positive 8 x and positive 5 x squared and x cubed. And there's my final answer at the bottom.
So when you're doing these, you have to make sure these columns line up correctly meaning these are all exes, these are all x squared, these are all x cubed. I couldn't have an x cubed in here because then I couldn't add them together.
So let's take a look at another example. Here we're going to look at horizontal versus vertical. The two methods combined and when to choose either one. Well if you just have two binomials multiplied. The horizontal method works. You can do that. There's no problem with that. So we can use horizontal here. Or like the problem we had before where you have three terms times two terms. If you have two terms down here, it's perfectly acceptable to use the horizontal method but if we're over here where we have three terms by three terms, you could use horizontal but it may get a little messy. The vertical method with this type of problem keeps everything organized. Let's go ahead and do this one with our vertical method. 4 times 1 is 4. 4 times 3x is positive 12 x. 4 times x squared is 4x squared. We're done with the 4, so let's move to the 2. When I take 2 times 1, I get 2x – where is that going to go? Is it going to go under here or under here or under here? Right? It's going to go under the 12 x because those are like terms. 2 x times 3 x is 6 x squared. 2x times x squared is 2x cubed. All right now when we try the x squared times all of these up here, we're going to skip this spot and this spot. Because x squared times 1 gives me x squared so what column does that have to go in? It has to go in the x squared column. X squared times 3x is 3x cubed. X squared times x squared is x to the 4th. So if we add straight down, we get a positive 4, positive 14x, positive 11x squared, positive 5x cubed and x to the fourth power. So there is my final answer in multiplying these two trinomials together.
Let's look at one more scenario and that is having this trinomial by this binomial. And I said before you could use horizontal method, that's perfectly ok. I am going to use vertical method with this because I want to show you when this scenario – what happens with this scenario. 7 times everything. 7 times 3 is positive 21, 7 times 3x is positive 21x and 7 times x squared is 7 x squared. Now you may look at this
and go WHOA! why is this x squared not over here. Well it doesn't matter about the top, about having
the top aligned. You want to have it in order. But it's ok if these columns don't align. You want the
columns underneath to align. So let's do our x squared times everything. X squared times 3 is 3x
squared. So where will that go down? Will it go under the 21, the 21x or the 7x squared? Correct, it
would go under the x squared because like I just said these columns need to be lined up. So I have 3x
squared. X squared times 3x is 3x cubed. X squared times x squared is x to the fourth power. Add them
all up, positive 21, positive 21x positive 10x squared, plus 3x cubed and x to the fourth power. So
there's our final scenario there. So always make sure that you leave columns for each type. For
instance, if we were missing an x here, I still would slide the 7x squared over because you may have an x
down there. Always make sure you have the correct columns.
