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Course: Class 9>Unit 7

Lesson 4: Polynomials 2.4

Special products of binomials: two variables

Sal finds the area of a square with side (6x-5y). Created by Sal Khan and Monterey Institute for Technology and Education.

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• I have (2^m + 1)^2 and asked to give product using Special Products. I assumed this equals 4^2m + 4^m + 1. But if I substitute 2 for m this does not check. (2^2 + 1)^2 = 25 but 4^2*2 + 4^2 + 1 = 49. No matter how I try to substitute for m it doesn't work out. How would you write it in special product?
• Use the formula (a+b)^2 = a^2 + 2ab + b^2
The answer should be (2^m)^2 + (2)2^m(1) + 1^2 = 2^2m + 2(2^m) + 1
substitute 2 into the equation: 2^4 + 2(2^2) + 1 = 16 + 8 + 1 = 25

Your only problem was oversimplifying the equation--remember that multiplying 2 raised to a power by 2 is not the same as 4 raised to that power. (Easy example: 4^2 =/= 2*2^2; 2*2^2 = 2^3. So when you were rewriting the equation, you could have written 2^2m + 2^(m+1) +1.) Your other problem was in your a^2 term. You don't both square the 2 and multiply m by two. You only do one or the other. Basically, you just need to practice your exponent rules a little bit more.
• I am having trouble with16t^2-4t
• 1st Step: Look for a GCF (Greatest Common Factor)
Between 16t^2 and 4t
The GCF is 4t, so factor out a 4t
2nd Step: Factor out the rest of the problem
• how about when you have fractions like (5x/10+2x/4)?
• Is it okay if the final answer is 36x^2+25y^2-60xy. ?
• Usually you put the xy term in the middle, but it depends on how picky your math teacher is.
• When expanding -(x-2)^2 should I distribute the -1 first then expand (-x+2)^2 or should I expand to
-(x^2-4x+4) then distribute my -1?
• I would do the latter, only because I find it easier to expand when the x is positive.

EDIT: Crap, I was wrong. You should do the latter, not because it's easier, but because it's right. By distributing the -1 into the binomial before expansion, you change the outcome.
• How do you multiply other powers?
- @Triathlife
• Like x^2 times x^2? If so, then the answer would be x^4.
• What Makes a Special Product Special? Please answer my question… I really need to know and Im very curious about it.
• There's nothing magic about it, what makes special products special is that they are very easy to solve, and they are easy to remember HOW to solve. As DeWain said, you won't end up using them very often in the real world, but the few times you do it'll be a pleasant surprise. Another thing I have noticed is, many problems of this type show up in textbooks and on tests because teachers want you to know this stuff. If you don't know it you can still solve the problems, but A) you may not get full credit if you were instructed to use the special products formulas, and B) it will take you much longer. If you are taking a test and spend 5 minutes on a problem that should've taken 30 seconds (maybe this happens several times), you may not even have time to complete the test. Also special products are fun:)
• I understand what Sal is doing with double distribution but why is it always the second term as in... (5x+6) (6x-12) he takes the (6x-12) or whatever would be there.... why not the first term? Is it so the line up perfectly or just preference?
• does it matter what order you put them in
• *yes, it matters a lot like; our answers will be
different for the following two orders;*
1. for the order(6x-5y)(6x-5y)
2. for the order(6x-5y)(5y-6x)
(1 vote)
• At Sal uses the term FOIL. What is FOIL?

Video transcript

- Find the area of a square with side (6x-5y). Let me draw our square and all of the sides of a square are going to have the same length, and they're telling us that the length for each of the sides which is the same for all of them is (6x-5y). So the height would be 6x-5y, and so would the width, 6x-5y, and if we wanted to find the area of the square we just have to multiply the width times the height. So the area for this square is just going to be the width, which is (6x-5y) times the height, times the height which is also (6x-5y), so we just have to multiply these two binomials. To do this, you could either do FOIL if you like memorizing things or you could just remember this is just applying the distributive property twice. So what we could do is distribute this entire magenta, (6x-5y), distribute it over each of these terms, in the yellow (6x-5y). If we do that, we will get this 6x times this entire (6x-5y), so (6x-5y), and then we have -5y, - 5y times once again, the entire magenta, (6x-5y). And what does this give us? So we have, we have 6x times 6x, so when I distributed just this, I'm now doing the distributive property for the second time, 6x times 6x is 36x squared, and then when I take 6x times -5y, I get 6 times -5 is -30, and then I have an x times y, -30xy. And then I want to take, I'm trying to introduce many colors here, so I have this -5y times this 6x right over here so -5 times 6 is -30. - 30 and I have a y and an x or an x and a y, and then finally I have my last distribution to do, let me do that maybe in white, I have -5y times another -5y, so the negative times a negative is a positive so it is positive, 5 times 5 is 25, y times y is y squared. And then we are almost done. Right over here, we could say, we can just add these two terms in the middle right over here, - 30xy-30xy is going to be -60xy. So you get 36x squared -60xy +25y squared. Now, there is a faster way to do this if you recognize. If you recognize that if I'm squaring a binomial, which is essentially what we're doing here, this is the exact same thing as 6x-5y squared. So you might recognize a pattern. If I have (a+b) squared, this is the same thing as (a+b) times (a+b) and if you were to multiply it out this exact same way we just did it here, the pattern here is it's a times a which is a squared, plus a times b, +ab, plus b times a, which is also +ab, we just switched the order, plus b squared, +b squared so this is equal to a squared +2ab +b squared. This is kind of the fast way to look, if you're squaring any binomial, it will be a+b squared, it will be a squared +2ab + b squared. And if you knew this ahead of time, then you could have just applied that to this squaring of the binomial right up here so let's do it that way as well. So if we 6x, (6x-5y) squared, we could just say well, this is going to be a squared. It's going to be a squared in which in this case is 6x squared +2ab, so that's +2 times a which is (6x) times b which is (-5y), - 5y +b squared, which is +(-5y), everything is squared. And then this will simplify too, 6x squared is 36x squared plus, actually it's going to be a negative here because it's going to be 2 times 6 is 12 times -5 is -60, we have x and a y, x and a y, and the -5y squared is +25y squared. So hopefully you saw multiple ways to do this, if you saw this pattern immediately, and if you knew this pattern immediately, you could just cut to the chase and go straight here, you wouldn't have to do distributive property twice, although, this will never be wrong.