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Multiplying binomials by polynomials: area model

Sal expresses the area of a rectangle whose height is y²-6y and width is 3y²-2y+1.

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  • hopper cool style avatar for user Bob The Zealot
    If there are monomials, binomials, trinomials, then are there quanomials or quinomials?
    (16 votes)
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  • blobby green style avatar for user marlon
    Why do we have to learn math that were never probally never use in the real world?
    (5 votes)
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    • mr pants pink style avatar for user Felix Lehmann
      For everyone who is asking the same question (i guess Marlon is way past this anyway)...
      1. If you plan on studying any STEM subject, you will need this. Period.
      2. Also it's quite possible that you will need it in any advanced form of business, banking or insurance work.
      3. Even if you will never encounter a wild polynomial of the 3rd degree, going through this helps you think in a different way, use your brain and train it to push itself.
      It's the same with historic dates, the fine arts, music, PE, languages. It all shapes your brain to be used in different ways.
      Think about it as crossfit for your head. I guess no one needs to do push ups, squats or ride a bike real fast in the real world, but it all helps to keep your body functioning at a high level and keeps you healthy.

      I know school sometimes lacks any real world application, but becoming smarter through training can help you make a living in the long run.

      And one more thing, don't believe everything you think right now. Don't fall into the trap to let your 14, 16, 22 year old self limit your future options. I am 38 right now and decided to study Artificial Intelligence, but if you had asked me 20 years ago, I would have told you, I will never work in any field which requires math. ;)
      (2 votes)
  • aqualine ultimate style avatar for user Yolie Reyes
    So the only reason the expression -6y & -2y can represent a length because there are two terms? If there were only one term either( -6 or -y) & (-2 or -y) would this logic be applicable? My instincts tell me that there needs to be another variable to mulitply by the negative to give us a positive. Am I correct in thinking this?
    (5 votes)
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    • leaf blue style avatar for user Fivety-five
      What I understood was that you are thinking that if you have only one term (like -6y), it would not be able to represent a distance. This is incorrect. I'll write the different ways to represent distance.

      -6 = Not able to represent distance because it is just a negative number by itself and you can't have negative distances.

      -y = Can represent a distance if y is also negative so if y was -3, it would represent -(-3) or 3.

      -6y = Also can represent distance, if y is also negative. If y was -3, it would represent -6(-3) or 18.

      -6*-y = Also can represent distance, if y is positive. If y was 3, it would be (-6)(-3) or 18.

      y^2 = If y is negative or positive, it would represent distance.

      y^3 = if y was positive, it could represent a distance.

      So, you don't need two terms to represent distance, there are just some assumptions you have to make if you ever come across a problem like this one.
      (4 votes)
  • starky sapling style avatar for user 𝘽𝘼𝙏𝙈𝘼𝙉
    QUESTION

    Is there an easier way to do this? like a simple formula? thanks! :D
    (2 votes)
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    • leaf orange style avatar for user A/V
      I accidentally posted a comment, but if any other people have the same question,

      I do not believe that there are any formulas related to the product of a binomial and a polynomial. Distributing and collecting like terms is the simplest way to find the product.
      (4 votes)
  • duskpin tree style avatar for user Panhamorakath You
    How do you multiply a binomial area model? and what is the definition of binomial in math?
    (1 vote)
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  • female robot grace style avatar for user 255432
    when sal is adding -18y3 and-2y3 why doesn't he add up the exponents so it equals -20y6 instead of -20y3?
    (1 vote)
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  • aqualine ultimate style avatar for user Iamtryingtoworkhardplayhard
    At , we knew that the two equations are the same but how do you factorize 3y^4 - 20y ^3 +13y^2 - 6y into (y^2 - 6y)(3y^2 -2y+1)?
    (2 votes)
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  • duskpin seed style avatar for user rsaintjusme
    Are we adding or multiplying the exponents? I'm confused.
    (2 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      When each product is calculated to get a term, the exponents are being added.

      Recall that when multiplying expressions using the same base, add the exponents.

      If you forget this rule, try a simple example such as x^2 * x^3 using the basic definition of exponents:
      x^2 * x^3 = (x*x)*(x*x*x) = x*x*x*x*x = x^5.

      Note that 2 and 3 are added, not multiplied, to get the 5.
      (1 vote)
  • aqualine ultimate style avatar for user Ari Garfield
    can this be done with other shapes?
    (0 votes)
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    • aqualine sapling style avatar for user Oreo
      Not necessarily. When you are doing area for a rectangle or square, you multiply length and width(A= l x w). If you did a different shape, say a triangle, you'd have to multiply your product by 1/2. All we are really doing is multiplying the terms and adding like terms at the end, so it makes the most sense to use rectangles or squares to show polynomials being multiplied together
      (4 votes)
  • marcimus purple style avatar for user Churro-Sama
    This was going way too fast, how do I SLOW it down?
    -Chuy
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      In the lower right portion of the video window, you should see a gear symbol. If you select it, there are options for changing the video speed.

      Also, remember that you can pause the video when you start to get confused. You can back if up a little bit or read the transcript to pick up words that you might have missed. Then, resume playing the video.
      (2 votes)

Video transcript

- [Voiceover] What I wanna do in this video is figure out multiple ways to express the area of the entire large rectangle, which we see is made up of these six smaller rectangles. So there's a couple of ways that we can do it. One way is, we can just multiply the height of this big rectangle times the width of this big rectangle. So what's its height? Well, from here to here, that distance is going to be y squared, and then from there to there, that distance is going to be negative 6y. And I know what you're thinking. How can my distance be negative 6y? Isn't a distance always positive? Well, even negative 6y can be positive if y is negative, so it's completely reasonable to say, well, this distance could be negative 6y. So the entire height right over here is going to be, it's going to be y squared minus 6y. Or you could do it as y squared plus this distance, which is negative 6y, y squared plus negative 6y, which is the same thing as y squared minus 6y. So that's the height of this big rectangle. What's its width? Well, the width is going to be the width of this purple rectangle, it's going to be 3y squared, plus the width of this yellow rectangle, which is negative 2y, and that can have a negative out here, the same logic why this could have a negative, why the negative 6y could have a negative, and then plus the width of the blue rectangle. And so if you add them all together, the width of the entire rectangle is going to be 3y squared minus 2y, minus 2y plus one. And just like that, this expression that I just wrote down will give us the area for the entire, the area for the entire big rectangle. Now, there's another way to do it, and a big clue was that we subdivided the big rectangle into these six smaller rectangles, and we have the dimensions for the six smaller rectangles. And so we could find the area for each of these, and then we could add them all together. So let's look at this first one. Height times width. The area of this purple rectangle is gonna be the height, y squared, times the width, which is 3y squared, which is going to be equal to, it's gonna be three, and then y squared times y squared is y to the fourth power. What's the area of this yellow rectangle? Height is y squared. It's going to be y squared times the width, times negative 2y, which is going to give us negative 2y to the third power. What about the blue one? Well, height times width, it's gonna be y squared times one, which, of course, is just going to be equal to y squared. Now, this green one, it's gonna be the height, which is now negative 6y, times the width, which is 3y squared, which is going to be equal to, let's see, negative six times three is negative 18, and then y times y squared is y to the third power. Now, the area of this gray rectangle is gonna be the height, which is negative 6y, times the width, which is negative 2y, which gets us negative six times negative two is positive 12, y times y is y squared. And then finally the area of this rectangle right over here, it's gonna be the height, which is negative 6y, times the width, which is just one, which is equal to negative 6y. And so if we want the area of this entire rectangle, we can just add up the areas of the smaller ones, so it's going to be equal to the three, it's going to be equal to the 3y to the fourth, 3y to the fourth, plus negative 2y to the third power. Let me write this in a color that corresponds to that. Negative 2y to the third power, plus y squared, plus y squared, minus 18y to the third power, so minus 18y to the third power, plus 12y squared. Let's write that in black. So plus 12y squared, and then last but not least, we have the minus 6y, minus 6y. So this is an expression for the area of the entire thing, but we can simplify it more. So let's see, we only have one fourth degree term, so I'll just rewrite that. So we have one fourth degree term, so I'll just rewrite that. 3y to the fourth power. Now, how many third degree terms do we have? We have negative 2y to the third power. We have negative 18y to the third power. So if we add these two together, how many y to the third powers do we have? Well, negative two plus negative 18 is negative 20, negative 20y to the third power. And then how many second degree terms do we have? Well, we have one y squared right over here, and then we have another, 12y squareds. You add those together, you're gonna have 13y squareds. And then finally we still need to subtract the 6y. And there you have it, another expression for the area of the entire rectangle. And the whole point of doing this is to realize that this up here and this down here are equivalent, and that the way that we multiply this actually corresponds to exactly how we found the areas of the smaller rectangles right over here. You would say y squared times 3y squared is 3y fourth. Y squared times negative 2y is negative 2y to the third power. Y squared times one is y squared, which is exactly what we did when we found the area of these rectangles in this, I guess you could say, in this top row. And then you would take the negative six, and you would say negative six times 3y squared is negative 18y to the third. Negative six times negative 2y is positive 12y squared. Negative 6y times one is negative 6y. And just to realize that this isn't just some type of voodoo that we're doing. It completely makes sense when you think about in terms of an area model like this.