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All right, this is about multiplying polynomials part three.
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And in the last video, we ended by multiplying a trinomial times a trinomial using a vertical
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format. And this is what it looked like. We had x squared minus 3x plus 2 times x squared
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minus 4x minus 6. So we did x squared times everything. And the first polynomial, then,
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and negative 4x times everything in the first polynomial, and then the negative 6 times
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everything in the first polynomial. And we get 9 terms, and we add all the like terms,
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and then, we end up with x to the fourth minus 7x cubed plus 8x squared plus 10x minus 12.
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Okay, so now I'm going to show you another method for keeping track of all these terms using a box method.
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So we can take a box, and what we're going to do is say, well, we end up multiplying every term in the first
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parenthesis times every term in the second parenthesis. So what we do is want to make a box,
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a 3x3 because this happens to have three terms in the first parenthesis, and three terms in the second.
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So it would be a 3x3. And we'll take the first polynomial, and write each term in the first polynomial
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when the top of a column. So we have x squared minus 3x plus 2. That's the first polynomial we could
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across the top, and the second polynomial we put along the right hand side. So that's x squared minus 4x and minus 6.
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Now, we have 9 multiplications, and wherever they meet, that's where you put them
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up. So I could do 2 times x squared, and that'll go right here, 2x squared, and negative 3x times x squared.
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That's going to be a negative 3x cubed, and we can do an x squared times x squared. That's an x to the fourth.
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It doesn't matter the order. I could have done x squared times x squared first, and put this in here.
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So let's do this going from left to right. x squared times negative 4x is negative 4x cubed, negative 3x times negative 4x is plus 12x squared,
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and 2 times negative 4x is negative 8x. And we still have to use last row. x squared times negative 6 is negative 6x squared.
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Negative 3x times negative 6 is plus 18x, and 2 times negative 6 is negative 12.
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Now, if you write your terms in descending order, like the x squared, the x term, and the constant, how we did in both of these,
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the cool thing is, along the diagonals will be the light terms. So x to the fourth, there's only 1x to the fourth term,
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but there are 2x cubed terms in there right here, so we just add their coefficients. Negative 4 and negative 3 is negative 7x cubed.
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Now, we've got 3 terms for the x squared terms. You have 2 plus 12 minus 6. So that's going to be 8.
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x squared. And then we have negative 8x and plus 18x. That's the x term plus 10x. It's good to put your sign in front of it, careful.
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And there's only one constant negative 12. So these 5 terms, x to the fourth minus 7x cubed plus 8x squared plus 10x minus 12,
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put them all together, because you don't want to leave it looking like that. This is just the method you use to get the answer.
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Now, we need to write the answer. It's x to the fourth minus 7x cubed plus 8x squared plus 10x minus 12.
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All right, let's see if that's the same thing we got doing it the previous way. So x to the fourth minus 7x cubed plus 8x squared plus 10x minus 12.
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I'm going to try to remember these coefficients. 1 negative 7, 8, 10, negative 12. Here it is. 1 negative 7, 8, 10, negative 12. Yes, I got the same answer.
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Sorry, it's hard to see it on the same page, but they are the same.
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Okay, put the video on pause and use the box method to multiply 2x squared minus x plus 4 times 3x plus 1.
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Now, in this problem, there's 3 terms in one parenthesis and 2 terms. So you're going to have a box with 3 terms, let's say, across the top and 2 along the side.
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You could have made a rectangle the other way. We'll do that next. But go ahead and put it on pause and fill in your numbers, add like terms, and write your final answer.
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Okay, you ready? Let's fill these in. We've got 6x cubed, negative 3x squared, 12x plus 4, negative x, 2x squared.
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So you may have written those 6 terms in, you know, in different orders you've felt them in, but they should look be all in the same boxes I have here.
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And then if we add the like terms here along the diagonals of the light terms, we've got 6x cubed, negative x squared, 11x, because remember this is a negative 1x in here, and a plus 4.
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So my final answer would be 6x cubed, minus x squared, plus 11x, plus 4. So I hope you got that answer as well.
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Notice that you could have written the box this way, where you put the 3x plus 1 across the top, and the 2x squared minus x plus 4 along the side.
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And you should end up with the same 6 multiplications. They end up looking a little bit different just because of the way I oriented the rectangle here.
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But you will still get exactly the same 6 multiplications. And when I add like terms, it should all work out. So let's do this.
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So I filled in the 6 numbers that go, the 6 terms that go in the box. And then if we add like terms here, we get 6x cubed, we get negative x squared, we get plus 11x, and plus 4.
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So I'm still getting my answer, 6x cubed, minus x squared, plus 11x, plus 4.
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And I think that's exactly what we got up above. Let's check it out.
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Yep, I got the same answer. So it doesn't matter which way you want to orient your box. I usually do it so that across the top is the one that has the most terms.
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So the first way I did it. Right here, takes up a little less space on your paper.
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So we've gone over two methods so far, a vertical format in the box method. So let's take two binomials, 2x minus 3 times 5x plus 9, and multiply using both methods.
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So we'll go ahead and put this on pause and try it, and then come back.
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Alright, let's do the vertical method first. 5x times 2x is 10x squared, and 5x times negative 3 is negative 15x. And then we move over. 9 times 2x is plus 18x, and 9 times negative 3 is negative 27.
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So that gives us 10x squared, plus 3x, minus 27. Alright, now let's do it in the box. 2x times 5x is 10x squared, negative 3 times 5x is negative 15x, negative 3 times 9 is negative 27, and 2x times 9 is plus 18x.
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Notice that the same four terms 10x squared minus 15x, see them right over here, and 18x minus 27, it just depends how you prefer to write it.
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That is what is so cool about math, not just one way to do it.
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You can be creative, so we have 10x squared, plus 3x, minus 27, doing these two different ways.
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And of course, you could switch the order. You could put the 5x plus 9 on top, and the 2x minus 3, you'll still get the same answer here.
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So you could switch in the box method, what you put across the top, and what you put along the side, you will still get the same answer. So four different ways of doing the same problem using just these two methods.
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Now, you're going to go on to the next video, and there's another way to do binomials.
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This is called multiplying two binomials, and there's a couple other methods that might be easier for you that most people use when they're binomials.
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But to do something where you've got like a trinomial, you would want to use either the vertical format or the box method.