-For the past two days we have been learning and gathering information on Simplifying Rational Expressions. Simplifying Rational Expressions is basically finding the easiest form of the equation by simplifying it. And Rational expressions are usual in their simplest form, if it's numerator and denominator have no common polynomial factors. We've also been going over how to find the domain and zeros of certain rational functions.
First of all:
What is a rational #?
A rational number would be numbers such as .4=4/10=2/5. It can be written as fractions w/ integers in the numerator & denominator.
It would not be a number for example like: √5/2
Rational Expressions: ---> polynomial/polynomial or polynomial over polynomial
Ex:
x^3-4x/x^3-4x^2+4x = x(x^2-4)/x(x^2-4x+4) =
Basically what you do is you simplify to the lowest possible terms possible. Then once you are done simplifying you cancel out the same values (that's what the crossed out black values stand for). And then you end with your final answer which will be in the simplest form.
Domain: Is the set of all x values that work in the expression.
Ex:
x^3-4x/x^3-4x^2+4x = x(x^2-4)/x(x^2-4x+4) = x(x+2)(x-2)/x(x-2)(x-2) = x+2/x-2
Domain: All Real #'s - {2}
Which basically means All numbers work in the place of x to make the equation set equal to 0 except for 2.
Zeros: All x value that makes the whole expression 0.
Equation:
x^3-4x/x^3-4x^2+4x = x(x^2-4)/x(x^2-4x+4) = x(x+2)(x-2)/x(x-2)(x-2) = x+2/x-2
x=-2--> zero -2+2/-2-2 = 0/-4 = 0
Basically what you're trying to find is the number you can substitute the x with to make the whole fraction equivalent to 0.
Now here are some example problems:
Equation: (x-1)(x+1)^2/(x+1)(x^2-1)
Solving:
Domain: All Real #'s
So basically what happened within this equation when you simplified was that you had all of the same common value as denominators and numerators. So therefore in result they canceled each other out and you end up with a result of 1.
Equation: 3x^2/6x^2-9x
Solving:
x(3x)/3x(2x-3) = x
What happened with this problem was it was simplified to simplest form then once there we canceled out the 3x from both of them leaving us with an answer of x/2x-3.
Tips:
- Be Careful with your simplifying (formulas on page 172 and 173)
- Make sure to cancel out the same integers
- Don't always suspect certain equations are simplified completely if it seems a bit harder then most problems
-http://www.youtube.com/watch?v=B4bVlDgHF5I
-http://www.youtube.com/watch?v=7Uos1ED3KHI
-If you didn't understand this concept before I hope this helped with the gist of it! For more explanation on this go to: Pages 218-219 in your book.
This was a well done and informative blog. It was very helpful how you defined all the terms (rational numbers etc.). The "end tips" we're nice because they made sure we got all the key information. The videos were a nice touch and reminding us of the factoring equations in the book was very nice. This was a very well done blog. Nice job!!
ReplyDeleteChristian, you've really got a knack for communicating math via a blog or written venue. Your first paragraph is a great overview and then you go into some more details, like what comprises a rational number and a rational expression. Your examples are clearly done and show all your steps. My only suggestion would be, in the future, to use something like Notability so the fractions and exponents are a bit easier to see. Finally, I like that you included tips, however I think you may mean "factors" instead of "integers" in the second one and I'm not sure what you mean in the third one. Overall, though, nicely done and the videos are a good, extra resource (the first guy does a decent example, but as I mentioned in Sebastian's post, he's a little painful to watch).
ReplyDeleteThis was a good blog. I liked how you divided it up so that in one section you explained domain and in the other you explained how to find the zeros. What could have been helpful is if you did out your examples in notability because sometimes it is hard to follow problems with fractions when they are typed out like this. I also liked how you had some tips for simplifying and finding zeros at the end of your blog.
ReplyDeleteChristian! This is a very thorough post that clearly illustrates the steps to simplifying rational expressions. It does this by clearly separating out each group in colours and by using different effects like using lines to cross things out. I really like how you gave example problems, and then clearly went through the steps on how to solve them by using the steps that you already listed. The tips section at the end also really helps sum up the whole blog and helpful videos! Great job!
ReplyDelete-Gail
This was a very good blog! It was very helpful how you mentioned a term, in color, and then explained it thoroughly and gave an example that clearly illustrated it. The problems at the end were good practice problems, and the end tips were very helpful reminders as to what to watch out for and what to remember as you're solving. Overall, this was a very thorough blog. Nice job!
ReplyDelete