Operations with rational expressions — Basic example
About this video
A SAT-style problem asks for the expression equal to a difference of two rational expressions. The instructor factors the quadratic denominator, finds the common denominator, rewrites each fraction over it, and combines the numerators to match one of the answer choices.
You'll be able to
- factor a quadratic denominator to find a common denominator.
- rewrite each rational expression over a common denominator by multiplying top and bottom.
- combine numerators and distribute a negative sign to simplify the result.
Key words
- Common denominator
- A shared denominator that lets fractions be added or subtracted; here it is x plus 8 times x minus 3.
- Rational expression
- A fraction whose numerator and denominator are polynomials.
- Factoring a quadratic
- Rewriting x squared plus 5x minus 24 as the product of two binomials, x plus 8 and x minus 3.
- Distributing a negative
- Multiplying each term inside parentheses by the minus sign, so minus 7 times x minus 8 becomes minus 7x plus 56.
Transcript
- [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.
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