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Into H Find the 5 nolulion. und decomposition L 2 Give your of 9() compnenIS 1 for r(t) 1 1 4 Ht /...

Question

Into H Find the 5 nolulion. und decomposition L 2 Give your of 9() compnenIS 1 for r(t) 1 1 4 Ht /

into H Find the 5 nolulion. und decomposition L 2 Give your of 9() compnenIS 1 for r(t) 1 1 4 Ht /



Answers

Find the missing polynomial in the denominator of $\frac{9 h+45}{h^{4}} \cdot \frac{h^{3}}{h(h-2)}$.

All right we have the function notation to find as H. Of our is equal to one over R. Plus H. Sorry R. Plus four. Um And all you have to do when you're asked to evaluate like H. Zero as you have to identify all the R's in the problem and there's only one and plug in zero. Now I would assume you can do that math in your head that zero plus four is four and just leave your answer like that. So I'll do the same thing with the next one with h. -3. I'm assuming that when you replace negative three and for our negative three plus four would give you one. Um And you could just rewrite one to find that one is one. So looking at -5 then really the same arithmetic all you have to do is plug in negative five plus four is negative one and one divide by negative one is negative one. So moving on to part D. What are you doing H. Of X squared. You're just replacing the R. And the problem with X squared and there's nothing you can do besides just rewrite it as X squared plus four. But you're doing the same thing as the other problems um with E. You can simplify a little bit because you're doing H. Of X squared plus one. Um So after you know plugging in X squared plus one and for our you can then add four in there And 1-plus 4 would give me five. And lastly if you did H. Of X squared plus one. Um Each of X. Squared and then add 1 to it. Um That's gonna be the same thing as what you had in letter D. So after you write that then you have to write plus one over here. And I would just leave my answer like that you could simplify if you got the same denominator, but I don't see much of a benefit to doing that.

All right, so once again will begin by distributing and combining like terms. So this time I'm going toe weight on subtracting this from both sides. I'm going to just distribute and see what happens and saying, Hero, distribute and see what happens So they have three h squared minus 12 is equal to five, each squared minus five age minus nine inch since the and squared for each square term on the left on the right is greater than the one of the left. I'm gonna subtract three h squared from each side and also need to move 12 by adding 12 to each side. And while I'm combining these like terms and moving, the 12 also combined these two like terms. So all of this becomes zero equals to H squared, minus 14 age plus 12 penalty five. All three terms fortunes by a common factor of two. So this still zero equals H squared minus seven age plus six. Then I can factor this train oatmeal so that I can use the zero product property just off to make a squared equals by H times each. To make six. You can either do six times one or two times three I used to, and three that I would get a combined total of five when I add them. Um, so I need to six and one any time that the middle term is bigger than the last term. And the first term is has has a coefficient of one. Um, I know that I'm going to use the The one times this number and this ambition never be more than one bigger than this. If it is bigger, Um, this could be bigger, much bigger than this if this has a coefficient of than one. But in this particular case, I know that I can use six times one and then six plus one makes seven to make positive. Here they have to be the same sign. So if it's negative here, they have to both be negative that we can say that if this is true, if this multiplied by this equal zero, then one of those factors has equal zero. So here I would add six to both sides and here I would add one to both sides. So h equals six. Sorry or what? And those are the two possible solutions that if you substituted six in tow. All the ages or you substitute one in tow. All the agents, the equation would be a true statement.

Okay, so we want to evaluate the falling or actually simplified a falling. So in our numerator, we can actually practice into a to H plus one. And then we wanted H minus five here. Actually, that's to be a plus. And we should have minus here since we went nine age to be positive. And this is all over on that. See, you have a four h and then we want it plus one here. Then it's minus. One would see. Does that work? We would get minus four. Actually, we can have to you. And to hear this is for a two inch. This is where? Age? My village. And then we want being up here minus 28 year and in plus one. And now let's cancel outs or like, tears. And then we see that we're left with H plus five all over to H minus one.

Passion assets to divide so we can rewrite this as to H squared minus nine, being divided into 10 age to the fourth minus six H cubes minus 49 age squared plus 27 h plus 19. And there go with polynomial Long division is to multiply something by two h squared to cancel out our other each terms, and so the first term we have to cancel out is 10 h to the fourth. So what times to each wear gives us 10 age to the fourth, and that would be five. Each squared, and now we take five each squared and we multiply it by minus nine, and we get minus 45 h squared. And in order to cancel out our 10 age squares we have or 10 H to the fourth, we have to multiply the green expression by negative one. And then we'll distribute this negative sign to the 45 so it becomes positive and minus 49. Plus, a positive 45 gives us a minus four h squared, and then we'll just bring down our other terms. And the next time we have to cancel out is minus six H cubes So what times to each squared gives us minus six h cubes. And that would be minus two H or minus Sorry, minus three each. And then we take minus three h and we multiply it by minus nine and we get positive 27 h and again to cancel out our minus six h cubes, we multiplied the blue expression by minus one. We'll distribute this mine assigned to the 27. So it becomes negative and are 27. Send that cancelling too. And now we'll bring down our remaining terms. And now the next time we have to cancel is four h squared. So what times to H squared gives us minus for each squared, and that would be minus two. So we get minus four h squared two times or minus two times minus nine gives us positive 18. And in order to cancel our our for each squares, we can't we multiply the red expression by minus one. Those will cancel will distribute this negative science of the 18 and we're left with the remainder of one. And whenever we have a remainder of one, we have to ray a fraction, with our denominator being our divisor. So we get a final answer of five H squared minus three H minus to plus one over to H Squared minus nine.


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