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In this video, we're going to discuss special products. We're going to begin with
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the product of two binomials, a plus b times a plus b, which can be written as a plus b squared.
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This product can be visualized geometrically using the area model for a square.
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In this square, this side length is a plus b and a plus b. So the sum of each of these four areas
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will give us the product of a plus b squared. The area of the square a plus b squared is the same
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as a plus b times a plus b. Now let's look at the area of each of these rectangles.
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In this top area, we have a side length of a and another side length of a. So this is a square. And
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the area of this square would be a times a or a squared. In this next shape, we have a rectangle.
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It has a side length of b and another side length of a, and so the area of this rectangle would be
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the product of these two sides. We will write that as a b which also could be written as b a.
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In this upper rectangle, we have a side length of a and a side length of b. So again we have a b.
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And in this bottom portion, we have a square with side length b and b, so this area is b squared.
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The area of the entire larger square a plus b times a plus b is the sum of these four areas. This gives us a squared.
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We have an a b here and an a b here. Adding those two like terms together will give us two a b.
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And we have this square, which is b squared. So the product of a plus b squared is a squared plus two a b plus b squared.
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This leads us to two identities, a plus b squared which is written here
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and also similarly we could write a minus b squared, and what would change here is the
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sign of the two a b. We get a squared minus two a b plus b squared for a minus b squared.
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Let's do some examples. In this first example, we have three x plus five squared.
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Using our formula, a plus b quantity squared, we know that this product will give us
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a squared plus two a b plus b squared. So this is a, and this is b. And a squared would be
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three x squared. Then we have two times the product of a and b plus b squared.
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And now we just have to simplify this expression. Three x squared will give us nine x squared.
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Two times three times five gives us thirty, and we're multiplying
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that times x so we have thirty x. And five squared is twenty-five.
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So the product three x plus five squared is nine x squared plus thirty x plus twenty five.
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In our next example, we have four m minus one squared. Notice the minus sign here. And so what
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changes in the formula that is different from a plus b squared is the a b term or the two a b term
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which will be negative. So we begin with a squared, which will be four m squared.
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Minus two a b, two times four m times one. Plus b squared, plus one squared.
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Simplifying this expression, we get sixteen m squared. Here we have negative two times four m times one.
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This will be a negative or minus eight m. And one squared is one, so plus one.
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Four m minus one squared is equal to sixteen m squared minus eight m plus one. And our last example is two times x
minus
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two y squared. We need to work the product of x minus two y squared first. So we'll keep our two on the
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outside of our grouping symbols, and we'll use our formula to simplify x minus two y squared.
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We begin with a squared, which will be x squared. Minus two times a times b,
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so that will be minus two times x times two y. And then plus b squared, which will give us plus two y
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squared. Let's simplify what are in these brackets. This gives us x squared minus. This gives us four x y,
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and then plus two y squared is plus four y squared. And finally, we'll distribute our two
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to clear the grouping symbols, giving us two x squared minus eight x y plus eight y squared.
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Another special product is for the sum and difference of the same two terms.
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In this example, we have two x plus one times two x minus one. Let's
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use the distributive property first to multiply this out and see what happens.
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If I distribute this first binomial to each of the two terms in the second binomial, we get two x
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times two x plus one minus one times two x plus one. Distributing here gives us four x squared plus two x.
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And distributing here gives us minus two x minus one. Now combining like terms, you can see that what happens is that
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our linear terms here, two x and negative two x, will cancel out, leaving us with four x squared minus one.
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Whenever we multiply a sum and difference of the same two terms,
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this pattern will continue to happen. This is called the difference of two squares.
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The reason why it's called that is that you can see that the product
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here leads us with a difference, a subtraction, of two perfect squares. The formula for the difference of two squares is
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a plus b times a minus b gives us a squared minus b squared, the difference of two squares.
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Let's look at a few examples. First we have three x plus five times three x minus five.
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Notice that we have the same two terms three x and five showing up in both
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of these binomials with a difference of signs. So we'll use our difference of two squares formula.
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This will give us a squared minus b squared, where a is three x, so we'll have three x squared,
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and b is five, so we'll have minus five squared. Simplifying here gives us nine x squared.
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And minus five squared gives us minus twenty-five, the difference of two squares. In our next example, we have
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ten a plus b times ten a minus b. Again, we'll use our difference of two squares formula.
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This will give us the first terms squared, or ten a squared, minus the second term squared,
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which is minus b squared. Ten a squared gives us one hundred a squared, and we have minus b
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squared. So our product is one hundred a squared minus b squared.
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And in our third example, we have x squared minus seven times x squared plus seven.
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We'll use our difference of two squares formula, and we'll multiply or square the first term,
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x squared squared, minus the second term squared. Remember our exponent rules when I
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have a power raised to a power I multiply. So this will give us x to the fourth minus forty-nine.