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- [Instructor] Which
expression is equivalent to the above difference? All right, this is gonna be fun. So we're adding these two, I guess we could say rational expressions. And whenever you add any
expressions like this, if you add any fractions, you would wanna have a common denominator. And the same thing is true over here. And so one thing we can do is
let me actually see if I can, actually, let me rewrite the whole thing. So three over x squared
plus five x minus 24 minus seven over x minus three. Now, instead of, I guess,
the brute force way is to just multiply the denominators and say, okay, that's gonna be a common multiple. But there might be a
cleaner way of doing that, especially if x minus three
is one of the factors of this. So let's see if this is. So let's see, negative
three times positive eight is gonna be negative 24. And the negative three plus
positive eight is positive five. So this actually can be factored
into x plus eight times, x plus eight times x minus three. This is what this thing is and if this step looks unfamiliar to you, I encourage you to watch the video on factoring quadratic expressions. But what's neat about this is now we say, okay, look, to have the
same denominator here, we just have to take this fraction and multiply the numerator and
denominator by x plus eight. So if we multiply the
denominator by x plus eight, then these two denominators
are equivalent. But I can't just multiply
only the denominator by x plus eight. I also have to multiply the
numerator times x plus eight. And so what does this simplify to? This is going to be equal to three over x plus eight times x minus three. Times x minus three. And then minus, and actually, let me distribute the seven. Minus, remember this
negative sign out front, so it's gonna be minus seven. Actually, let me just
write it this way first. Minus seven x plus 56 over x plus eight times x minus three. If I was really under
time pressure on the SAT, I wouldn't probably do all of these steps. I would try to get to the chase, cut to the chase a little bit faster. But this is hopefully so it
helps us understand things. And so this is going to be equal to. We have our common denominator. X plus eight times x minus three. So it's going to be three. And we can distribute this negative sign. Minus seven x minus 56. So it's going to be equal to, let's see, we can write negative seven x. And then three minus 56 is
minus 53 or negative 53. And x plus eight times x minus three, we already established, is the same thing as x squared plus five x minus 24. And that is exactly, that's exactly this
choice right over there.