We’re starting to see a difference of squares emerge…
Multiplying binomials. FOILing. Whatever you call it, and however bad we want it, there’s no real shortcut. So why does (x + 5)2 ≠ x2 + 25? Let’s take a look:
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Above is a representation of (x + 5)2. We can see along the top edge “x 1 1 1 1 1”, representing x + 5. Whenever we square something, we multiply it by itself, so we see the same x + 5 along the left edge. Since (x + 5)2 = (x + 5) times (x + 5), let’s multiply to find the area of each colored region:
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If we put all the pieces together, we get:
(x + 5)2 = x2 + 10x + 25
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When we say that (x + 5)2 = x2 + 25, we miss out on all of those little blue 1x’s. Multiplying two expressions together will always give us an area. For example, a rectangle with length 5 and width 3 will have an area of 15. Multiplying two binomials together, like we did above with (x + 5)(x + 5), usually yields a trinomial. I say usually because there is one case when this is not true…
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Let’s multiply (x + 5)(x – 5). A great way to do this is with the Box Method:
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Above, we see (x + 5) along the top of the Box and (x – 5) along the left. If we multiply these two binomials together:
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and then combine like terms, we get: x2 – 25. Since both x2 and 25 are square numbers, and they are being subtracted, we literally have a difference of squares. There is no middle term because the +5x and the -5x cancel each other out.
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To see how this problem translates into areas like our first example (x + 5)(x + 5), let’s start at the end and work our way back to the beginning….
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Here we see two squares: one is green and one is white. The white one is being subtracted (difference) from the green one.
Since “difference” means subtract in the language of Math, we quite literally have a difference of squares. Above, we see 52 being subtracted from x2. To make things more interesting, let’s overlap the regions:
Because the green shape is pretty lopsided now, let’s draw some dotted lines to think about the green shape in terms of three nice, regular shapes:
And now let’s multiply to find the areas of each of the nice, regular shapes:
If we simplify each of the white expressions, we get:
5(x – 5) = 5x – 25
5(x – 5) = 5x – 25
(x – 5)(x – 5) = x2 – 5x – 5x + 25 = x2 – 10x + 25
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And then if we add them up:
(5x – 25) + (5x – 25) + (x2 – 10x + 25) = x2 – 25 It’s a difference of squares!
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But can we express this x2 – 25 as the product of two expressions, like we did with x2 + 10x + 25 –>(x + 5)(x + 5)? When we ask this question, we’re asking if we can go backwards; we’re asking if we can factor the expression to find out where it originally came from.
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In the first example, x2 + 10x + 25 factored to (x + 5)(x + 5). Can we do the same with x2 – 25?
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Let’s go back to our overlapped picture to find out:
Maybe if we break up the green region:
And begin to rearrange the pieces, first sliding one rectangle up:
and then chopping that bottom part, rotating it 90° and putting it on the left:
We made a rectangle! And what are its dimensions?
(x + 5)(x – 5)!
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So x2 – 25 came from (x + 5)(x – 5). In this situation we didn’t get a middle x term when we multiplied the two binomial expressions together. Instead, we got a difference of squares, which makes sense since that’s where we started!
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Here’s a video that shows why (a + b)2 ≠ a2 + b:
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Contact this blog’s author at shanadonohue@gmail.com.
