Simplifying rational expressions combines everything learned about factoring common factors and polynomials. When simplifying rational functions, factor the numerator and denominator into terms multiplying each other and look for equivalents of one (something divided by itself). Include parenthesis around any expression with a "+" or "-" and if all terms cancel in the numerator, there is still a one there.
Adding and subtracting polynomials is really just a fancy way of combining like terms, one thing you have to be careful with subtracting though of course is remembering that the minus sign applies to every term every polynomial. But first let's look at some different ways you can organize your work for adding and subtracting polynomials. One way some students like to do is to arrange their work vertically, another way is to group each term according to it's degree. Let's do the vertically one first, you're asked to simplify this polynomial plus that polynomial notice these are actually 2 trinomials, trinomials because they've got 3 terms and don't be confused when you parenthesis in your homework. A lot of times we see parenthesis and we think okay parenthesis means probably multiplying but be careful this parentesy term is being added to that parenthesy term. It would multiplying if there wasn't that plus sign now this would mean multiplying. In our case there was a plus sign we know that we're adding. Okay let me show you how vertically these could be arranged. If I were to write these 2 trinomials on top of each other it might help a visual learner to see how to add them. For example, here's my first trinomial 3x squared plus 6x plus 7 and then to that I'm going to add 3x squared take away 9x take away 8. Now I'm just going to add vertically 3x squared plus 3x squared is, whoops not 9, we're adding. 6x squared, 6x take away or plus negative 9x is take away 3x's and then 7 plus negative 8 is negative 1, that's my answer right there. This is one way you could do this problem by writing it vertically, just be really careful when you're lining up things that you line up your x squareds with your x squareds, your regular x's with your regular x's. Also if this had been a subtraction problem be really careful when you write it that you remember that this minus sign would apply to each of these terms. Let me show you another way to do this problem, notice this in blue here is the exact same problem as this guys here. I'm going to do this exact same problem another way and here's how the other way works. When you're grouping these problems according or grouping these terms according to degree remember degree means the exponent. So if I look I'm going to be looking for all my x squarederes I have 3x squareds and then 3 more x squared so altogether that is 6x squared that's going to be the first part of my answer then I'm looking for my regular x terms I have 6x here take away 9x so altogether that's negative 3x. And then last but not least I want to look for my constant terms or the terms without any x's. Here's 7 take away 8, 7 take away 8 is negative 1 so that's my constant term and here is my final answer. You can see this is the same problem so I got the same answer that's good, this way it requires a little bit more writing so to be honest most of my students use this method. Just be really careful when you're doing this method that you account for every single term in both your polynomials. You don't want to leave anybody out. I personally like to use color or different markings like you see I did brown squigglies on my x squared terms I circled my regular x's in a different color and I did underlinings for my constant terms. That just helps me as a visual person to not only show that every single term has been accounted for but also to see which terms are getting grouped together because they have the same degree. So when you guys see these kinds of problems adding and subtracting polynomials it's up to you if you chose to rewrite it vertically or if you like to do this grouping according to degree either way it's fine just make sure you're really precise with your positive and negative signs.
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