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Special products of the form (x+a)(x-a)

Sal introduces difference of squares expressions. For example, (x+3)(x-3) is expanded as x²-9.

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  • aqualine seed style avatar for user manelee
    cant you just used the FOIL method?
    (14 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Another thing is that the FOIL method is generally not useful when multiplying something other than two binomials (that is, multiplying either three or more polynomials or multiplying polynomials with three or more terms).

      However, multiple applications of the distributive property work for finding any product of any polynomials. This ultimately means multiplying each term of each polynomial (using all the polynomials to be multiplied) and then adding the products.

      Have a blessed, wonderful day!
      (20 votes)
  • duskpin ultimate style avatar for user Saesha
    Can you distribute it like: x(x-3) 3(x-3) instead of x(x+3) -3(x+3)
    (7 votes)
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    • stelly blue style avatar for user Kim Seidel
      Almost... You would need to have: x(x-3) + 3(x-3)
      Your version is missing the plus sign in the middle.
      Without it, you have an expression that is "3x (x-3)^2" which will create a completely different result that what you want.
      Hope this helps.
      (13 votes)
  • blobby green style avatar for user Alina vang
    when will we ever use this IRL please It's a genuine question
    (6 votes)
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  • male robot johnny style avatar for user Dominique Saunders
    It is common to type x squared or x to the power 2 with a 'carrot.'
    x^2?
    (5 votes)
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  • blobby green style avatar for user Lananh Le
    what are the solutions to x(x-a)-b(a+b)=0? Please explain in detail. TY.
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      In its current form, it appears not to be factorable because for the 2 terms, the contents of the parentheses don't match. So, there is no common factor.

      Distribute to eliminate the parentheses: x^2-ax-ab-b^2=0
      Use commutative property to rearrange the terms. This may take a while to find the right sequence. I did:
      x^2-b^2-ax-ab=0

      Then group by pairs of terms and try to factor.
      x^2-b^2 = (x-b)(x+b)
      -ax-ab = -a(x+b)
      So, not it looks like: (x-b)(x+b)-a(x+b)=0

      Now the 2 terms have a matching factor of (x+b). Factor it out to get:
      (x+b)(x-b-a) = 0

      You can then use the zero product rule to split the factors and solve for the desired variable.
      x+b=0 and x-b-a=0

      Note: You didn't specify which variable you are solving for. So, I can't go further. I assume you were told to solve for "x" or solve for "b" as these occur in both factors.

      You should be able to take it from here.
      Comment back if you have questions.
      (4 votes)
  • duskpin ultimate style avatar for user Sophia
    Why did Sal distribute (x-3) out to (x+3) and not vise versa?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Actually, Sal did distribute the (x+3). If you notice, the (x+3) is what is getting multiplied with the "x" and the "-3" after the distribution. If he had distributed the (x-3), the result would have been:
      x(x-3) + 3(x-3)

      Note: either method is acceptable because of the commutative property of multiplication.

      Hope this helps.
      (2 votes)
  • duskpin tree style avatar for user Valmue
    how do you solve (x+1)^3? My answer keeps besoming x^3+3x+1, but the book I use says the right answer is x^3+3x^2+3x+1.

    How do I get to the 3x^2?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      It looks like you tried to apply the technique for squaring a binomial to cube the binomial. It doesn't work to cube a binomial.

      Think about how you cube any number...
      If you were to do: 5^3, you would do 5*5*5 = 25*5 = 125
      You need to do the same process.
      -- Multiply 2 factors: (x+1)(x+1) = x^2+2x+1
      -- Then, take the result and multiply it with the 3rd factor
      (x+1)(x^2+2x+1) = x^3+2x^2+x+x^2+2x+1 = x^3+3x^2+3x+1

      Hope this helps.
      (2 votes)
  • blobby green style avatar for user Imanga Lufungulo
    Why is it ax instead of xa?
    (1 vote)
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  • leafers sapling style avatar for user angelsantos4214
    Hello,
    Why x (x+3) -3 (x+3) is equal to x^ +3x-3x-9?
    I got confused a little bit about why the "3" have "x" on it?
    Like how the "3" got "x" on it?
    Like this "3x"?

    But I understand a lot of this, except "3x"!
    (1 vote)
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  • aqualine ultimate style avatar for user Zahin Majid
    () why did we pick (x+a)to work with and not (x-a)
    (1 vote)
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Video transcript

- [Voiceover] Let's see if we can figure out what x plus three times x minus three is, and I encourage you to pause the video and see if you can work this out. Well, one way to tackle it is the way that we've always tackled it when we multiply binomials, is just apply the distributive property twice. So first we can take this entire yellow x plus three and multiply times each of these two terms. So first we can multiply it times this x. So that's going to be x times x plus three. And then we are going to multiply it times, we can say, this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again. And then we apply the distributive property one more time. Where we take this magenta x and we distribute it across this x plus three so x times x is x squared, x times three is three x, and then we do it on this side. Negative three times x is negative three x and negative three times three is negative nine. And what does this simplify to? Well, we're gonna get x squared, and we have three x and minus three x so these two characters cancel out, and we are just left with x squared minus nine. And you might see a little pattern here, notice I added three and then I subtracted three and I got this, I got the x squared and then if you take three and multiply it by negative three, you are going to get a negative nine. And notice, the middle terms canceled out. And one thing you might ask is, well, will that always be the case, if we add a number and we subtract that same number like that? And we could try it out. Let's talk in general terms. So if we, instead of doing x plus three times x minus three, we could write this same thing as, instead of three, let's just say you have x plus a times x minus a. And I encourage you to pause this video and work it all out, just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto the x and the negative a. So x plus a times x, or we could say x times x plus a, that's going to be x times x plus a, and then we're going to have minus a, or this negative a, times x plus a. So minus, and then we're gonna have this minus a times x, plus a. Notice, all I did is I distributed this yellow, I distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And now we can apply the distributive property again. X times x is x squared, x times a is ax, and then we get negative a times x is negative ax, and then negative a times a, is negative a squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out, and what are we going to be left with? We are going to be left with x squared minus a squared. And you can view this as a special case. When you have something, x plus something, times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. And we could use it to quickly figure out the products of other binomials that fit this pattern here. So if I were to say, quick, what is x plus 10, times x minus 10? Well, you could say, all right this fits the pattern, it's x plus a times x minus a, so it's going to be x squared minus a squared. If a is 10, a squared is going to be 100. So you can do it really quick once you recognize the pattern.