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Structure in expressions — Harder example

Video transcript

- Which of the following is equivalent to the above expression? So I have this kind of hairy-looking expression here, and I have a bunch of choices. And at least two of the choices have me factoring this expression in some way. And when you look at this expression with a factoring lens, as hairy as it looks, you realize that this actually is a difference of squares. That this thing could be rewritten as, let me write it this way, as the product of, actually let me write it this way first. This could be rewritten as, let me do this in a color you can see. This part could be rewritten as a times one over 2x minus y and all of that squared, because notice if you have a times a square, that's the same thing as a squared times this thing squared, which is exactly what you have there. And then minus one squared. Right? One is the same thing as one-squared. And when you write it this way, it becomes very clear that this is a difference of squares. Or even better, we can write it like this. Because a times one over 2x minus y, that's the same thing as a over 2x minus y, so it's that squared minus one-squared. And so let's factor this out. This is going to be equal to, actually let me write it. So it's going to be the product of two expressions. So that's the first one, that's the second one. And this is once again, if I have the difference squared, something squared minus something else squared, it's going to be, this is going to be a over 2x minus y, plus this thing squared, plus one. Plus one times a over 2x minus y, minus one. Minus. Minus one. And once again, nothing fancy here. This is just the difference of squares. You might have remembered from your algebra class, if you have x squared minus y squared that that's the same thing as x plus y times x minus y. Just in this example, x is a fairly hairy expression, and y is just the number one. But it's the exact same thing that's going on here. And if you look at the choices, and if the term difference of squares seems foreign to you, search for it on Khan Academy, it's a good place to review. But if we look at the choices, this is exactly what we have here. They just swapped the order of these two expressions being multiplied. A over 2x minus y minus one. times a over 2x minus y plus one. So that's that choice right over there.