Simplifying rational expressions
Simplifying Rational Expressions 3 U11_L1_T1_we3 Simplifying Rational Expressions 3
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- Simplify the rational expression
- and state the domain.
- Let's see if we can start with the domain part of the
- question, if we can start with stating the domain.
- Now, the domain is the set of all of the x values that you
- can legitimately input into this if you view this as a
- function, if you said this is f of x is equal to that.
- The domain is a set of all x values that you could input
- into this function and get something that is
- well-defined.
- The one x value that would make this undefined is the x
- value that would make the denominator equal 0-- the x
- value that would make that equal 0.
- So when does that happen?
- Six minus x is equal to 0.
- Let's add x to both sides.
- We get 6 is equal to x, so the domain of this function is
- equal to the set of all real numbers except 6.
- So, x could be all real numbers except 6, because if x
- is 6 then you're dividing by 0, and then this
- expression is undefined.
- We've stated the domain, now let's do the simplifying the
- rational expression.
- Let me rewrite it over here.
- We have x squared minus 36 over 6 minus x.
- Now, this might jump out at you immediately, as it's that
- special type of binomial.
- It's of the form a squared minus b squared, and we've
- seen this multiple times.
- This is equivalent to a plus b times a minus b.
- And in this case, a is x and b is 6.
- This top expression right here can be factored as x plus 6
- times x minus 6, all of that over 6 minus x.
- Now, at first you might say, I have a x minus 6
- and a 6 minus x.
- Those aren't quite equal, but what maybe will jump out at
- you is that these are the negatives of each other.
- Try it out.
- Let's multiply by negative 1 and then by negative 1 again.
- Think of it that way.
- If I multiply by negative 1 times negative 1, obviously,
- I'm just multiplying the numerator by 1, so I'm not in
- any way changing the numerator.
- What happens if we just multiply the x minus 6 by that
- first negative 1?
- What happens to that x minus 6?
- Let me rewrite the whole expression.
- We have x plus 6, and I'm going to distribute this
- negative 1.
- If I distribute the negative 1, I have negative 1 times x
- is negative x.
- Negative 1 times negative 6 is plus 6.
- And then I have a negative 1 out here.
- I have a negative 1 times negative 1, and all of that is
- over 6 minus x.
- Now, negative plus 6.
- This is the exact same thing as 6 minus x if you just
- rearrange the two terms. Negative x plus 6 is the same
- thing as 6 plus negative x, or 6 minus x.
- Now you could cancel them out.
- 6 minus x divided by 6 minus x, and all you're left with is
- a negative 1-- I'll write it out front-- times x plus 6.
- If you want, you can distribute it and you would
- get negative x minus 6.
- That's the simplified rational expression.
- In general, you don't have to go through this exercise,
- multiplying by a negative 1 and a negative 1.
- But you should always be able recognize that if you have a
- minus b over b minus a that that is equal to negative 1.
- Or think of it this way: a minus b is equal to the
- negative of b minus a.
- If you distribute this negative sign, you get
- negative b plus a, which is exactly what
- this is over here.
- We're all done.
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