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- Let's do a few examples adding rational expressions.
- So let's say we had negative x squared over x squared
- minus 7x plus 6.
- And to that I want to add plus x minus 4.
- So to some degree what I want to do in this first example is
- kind of our algebraic long division or polynomial
- division in reverse.
- I want to put this x minus 4 in the numerator with x
- squared minus 7x plus 6 in the denominator.
- The easiest way to do that is to just literally multiply x
- minus 4 times x squared minus 7x plus 6 over x squared
- minus 7x plus 6.
- I'm not changing the number, I'm multiplying it by 1.
- So if we do that what do we get?
- Well we just have to multiply this numerator right here, and
- we can do that with algebraic multiplication.
- So if we have x squared minus 7x plus 6 times x minus 4, we
- can multiply these two.
- 6 times negative 4 is negative 24.
- Negative 4 times negative 7x is positive 28x.
- Negative 4 times x squared is negative 4x squared.
- And if x times 6 is 6x, x times negative 7x is negative
- 7x squared.
- x times x squared is x to the third.
- So the new numerator is going to be x to the third minus,
- this is 11x squared plus 34x minus 24.
- So now I can rewrite this-- let me copy and paste this--
- this is going to be equal to that.
- Plus I can rewrite this thing now as being over a
- denominator of x squared minus 7x plus 6.
- And our numerator, x minus 4 times this thing over here, we
- get this whole thing.
- We multiplied them out. x to the third minus 11x squared
- plus 34x minus 24.
- And now, since we have the exact same denominator we can
- add these two rational expressions.
- So if you add them, you literally can just say well we
- have the common denominator, so our common denominator is x
- squared minus 7x plus 6.
- That was the whole point of multiplying this expression
- over this expression so that we could get the common
- denominator.
- And now we just add the numerator.
- So this is going to add to the x squared side, so you're
- still going to have an x to the third.
- And then negative or minus 11x squared minus another x
- squared is minus 12x squared plus 34x minus 24.
- And we are done.
- Now we could have checked whether we could have factored
- this, but this just factors into x minus 6
- times x minus 1.
- So it really doesn't factor with the minus 4 in any
- meaningful way, so there really isn't any further
- simplification that we can do for this problem.
- Hopefully you enjoyed that.