find-the-interpolating-polynomial-pta_0a_1-ta_2-t2-for-the-data-112215316-that-is-find-a_0-a_1-and-a

Question

Find the interpolating polynomial $p(t)=a_{0}+a_{1} t+a_{2} t^{2}$ for the data $(1,12),(2,15),(3,16) .$ That is, find $a_{0}, a_{1},$ and $a_{2}$ such that
$a_{0}+a_{1}(1)+a_{2}(1)^{2}=12$
$a_{0}+a_{1}(2)+a_{2}(2)^{2}=15$
$a_{0}+a_{1}(3)+a_{2}(3)^{2}=16$

Michael Jacobsen · Accepted Answer

for this exercise, we&#x27;re going to be solving for the degree to polynomial p a t such that this polynomial is going to pass through these three given points. Let&#x27;s start by using the function notation as follows. We know that p of one must be equal to 12 p of two. Must be equal to 15 and p evaluate at three must give us back 16. Then, if we look at the left hand side, P of one is just placing a one here and here, for example, so it would become the equation. A not plus a one times one for T plus a two times one for T squared is equal to the right hand side. 12. Let&#x27;s do this again by placing to here and here and setting it equal to 15 all together, we would obtain the second equation that is a not plus two times. A one two is then going to be squared. Soul go plus four times a two, and that&#x27;s going to be equal to 15 likewise, replacing three into pft and we get the equation that a not plus 3 +81 plus 9 +82 must be equal to 16. Now, at this stage we have a system of three linear equations in three unknowns. So if we air consistent, then will be able to determine a solution for Pft. Let&#x27;s next take this to an augmented matrix for this matrix we see here. This portion has been augment has been placed here, and the right hand side has been augmented with the 12 15 and 16 Let&#x27;s begin role reducing to see if we have a solution. So for the first reduction step, I&#x27;m going to take negative one times. Row one added to row to replacing row two. So first we have 111 12 in Row two will become zero one, three and three now for Row three would do the same operation multiply row one by a negative one added to row three and record the result here. The result is going to be zero to eight and four. Now for the next step in solving this system, let&#x27;s pivot off this entry here. We&#x27;re going to take to row operations like before, But first, let me copy that middle row, which is 0133 Now. Our goal is to eliminate this entry above next. So multiply wrote to that. We see here by negative one at it to row one and place the result here we&#x27;ll obtain one zero negative two and a nine. Likewise, to eliminate this entry multiply row to buy negative to And at the result to row three, we&#x27;ll obtain zero 02 and negative too. Now, we&#x27;re just about there for the next operation. Let me take an extra step where we divide this entry by two, thereby obtaining an entry of one or record. The result down here will have 001 negative one. And you can do row operations at this step filling out these entries here. I feel like that&#x27;s a lot of work to do at once. So let me just copy in the remainder of the Matrix. So now we have two objectives like before pivoting off of this entry here. We&#x27;re going to eliminate first this entry so we can say that this is robe equivalents to the following first. Copy that last row 001 negative one. Then multiply Row three by negative three and add the results of Row two will have 01 zero by design and then six here for the next operation. We want to get rid of this negative, too. So if we multiply row to buy positive too, add it to row one will achieved. That result will have 10 zero here, and a seven at this stage were ready to record what we were able to solve for recall that the unknowns are a not a one and a two. So by our form of the system, a not is equal to seven. A one is equal to six kind of missed their six and A to this is a zero here is equal to negative one. So that tells us that the required polynomial is p f. T equals a knot, which is seven plus a one, which is six times t plus a two, which is negative one times t squared. So this is a required polynomial that goes through all three points

Grace Muhihu · Answer

Okay, so if we start plugging in negative two and 40 so we&#x27;ll have native two squared minus six times native to minus one. So if we do this will get 15. And then if we plug in 1/4 so we&#x27;ll have 1/4 squared minus six times 1/4 minus want. So if we do that, we&#x27;ll get a result of negative 39/16 So these two are gonna be our results.

Edwin Bakalemwa · Answer

right. So in this question, you are give them a polynomial, uh, t. And you&#x27;re told that he has ah, riel coefficients. Well, you&#x27;re also told that you has degree three. And you told to assume that the lead coefficient p is one and, uh, father, your tour that zero is p our three, and, uh, they get too high. So the question is, find pulling nam u T being Now we know that the complex views, but really no meal with real coefficients. School is a car. Who cares? Code contradict pairs. Okay, so in this case, um, you know, um, negative to I please. Zero. As it has been said, Well, to I must be zero, because negative, too high into I are complex confidence. You. So Yeah. Now we have three, uh, zeros and so weaken. Like TX, we can like p x eyes now, because really is a zero, then X minus three is a factor off key. And because I got to I he&#x27;s another Zino. They&#x27;re expressed to High is another factor or feet. And we have just figured out that too. I he&#x27;s a zero. So thanks. Minus two I He&#x27;s another factor he&#x27;s another factor full. Oh, hey. So we can simply kindness by writing this and then we can much by the last two factors, they look like a difference of squares. So I can register X minus two. I squared. Cool. So these these eggs minus three. And this is sorry, Mrs. Exit script, PeopleSoft squealers. So here you come, X plus four squared plus four. And, uh, if you might buy Alex the brand mark about the brackets here you have ex cute minus three exits squeal. And then, uh, plus works minus 12. So this is the polynomial.

Edwin Bakalemwa · Answer

All right, so in this question, you have been given a Pelino me O p You&#x27;ve been told that he has really confusions, okay? And you&#x27;re also told that agree? He&#x27;s for who wants have been told that the leading coefficient Oh, me. He&#x27;s one. Okay. Also, we have been given that zeros off. All right, three and to I. So the question is find pulling on you. P you so now here? Because he has four degrees. Sorry, he has a degree for but it must have for zeros. And as you can see here, only Tuesday wasn&#x27;t been given. So we need to find to add a zeros. And then we can use zeros to build up beat or binomial. He so here. Negative. Sorry to I he&#x27;s a zero. Okay. And since complex zeros must occur in country in pairs, then negative to I. He&#x27;s also is zero. Okay, well, the reason being corn to get pants. Okay. And also here, as we have noticed before, P has agreed for and so found unable to identifying just early three zeros, then we need to find a 40 now in this case, three must definitely be repeating a zero off T. These are featured. Zero open. Okay, Now that we have all four zeros, we can go ahead and viewed the problem your p x by using the contra fear. Okay, so the first factor here to concede a would be X minus three. Yeah, well, the reason being the extra three is a zero. The next one would be X minus two. I express toe I and finally, because three Is there a pigeon zero? Then we have X minus. Okay. So we can rearrange this. Look, Wake X men of three square and then X minus two. I hear express to buy. So x Ministry squid is X squared minus six x last time. Andi. I can see that the last the last time was to buy No muse. A difference of squares Lincoln rightness as excess cleared minus to my squared. Okay, so he sees X squared When speaks x Is this excess weird? Last four. My blood is out with her export for minus six. Exe Cute. Last 13. Excess weird by most 24 x plus the souks. Okay, so this is the polling on U P. As required

Edwin Bakalemwa · Answer

But so in this question, you have been doing a polynomial t and your toward the P has really acquisitions. And you&#x27;re also told that he has a degree for on, uh, we have been asking to assume that the need Confucian off he is one. And also, you have been given the roots off p to be some sort of zeroes off key to be Are you negative too? And, um, we get to three. So the question is, find the polynomial p x. Okay, So since, uh, complex dealers must all current and go in confident pairs and since negative three Hi, he&#x27;s a single then three I He&#x27;s also zero because later to C I c I for my conjugated pair and also here, since he has decree four, then negative too. Must be a repeated roots. So now do really we can see that, uh, king assess the zeros has negative tunes. Make it industry I three i in negative two zeros. Okay, so he X therefore is equal to X plus two because negative two is around zero and also X minus two. Sorry, plus two. Because this is a repeated harder route and also we have X, uh, plus three pie as well as X minus to B. Okay, so the multiplies of the bracket you have Thanks. Squid was for X last four. And this look like, uh, defensive squares. So this is the sinking of X squared minus three, right? Squeamish, right? Minus three high. He&#x27;s so he sees it&#x27;s it&#x27;s weird was for X plus four bracket census plan last month. Okay, so if you much by at the brackets for this, you get next. Four us for exe Cute West team. Thanks. Is cleared first in his six. Thanks. Last 36. So this is Felix Clinton and is for

Edwin Bakalemwa · Answer

Okay, So in this question, you have been given a Pelino. Me, O P. And I have been told that he has real coefficients. You have also been told that he has degree for your boss. Have been told that they lied. Condition toe t he&#x27;s one. And also that the zeros Jeannie are negative. One and one minus three. So the question is, find the polynomial Jane. So now here API has four degrees. Sorry. Big apartment. Gonna freeze that again because he has degree for it. Must have four zeros and only two zeros have been given. So we need to find two of zeros. And then we can use a factor fearing to do the polynomial P. Now, I want you to note here that ah, one minus three, I use a zero. Okay. And since complex zeros muscle can contribute pains, then, um, one plus three. I is also zero. Okay. And I can see the reason for these things. Confident can can you? So now we have three zeros. It&#x27;s or it&#x27;s one of this one. And because four Sorry, because he has a degree for and it need to have four zeroes. Then we can say most definitely that negative one. Here, use a repeated zero kay. Now, with all these four zeros and using factor feeling putting on you, we can go ahead and build the polynomial p x, my raking it as a product off off the factors. So the first factor here we can consider he&#x27;s X plus one because negative one is a zero. The next one would be X minus on one ministry I The next factor B X minus one plus three I even last year, as we just mentioned. Ah x plus one because negative land is at a peak age zeal, Jerry. So this can be rearranged to look like thanks less when squared the next bracket and one of its key pewter negative sign here. And this gets to be X minus one plus three I and put the bracket here. And for the next one we distribute negative one. So minus one minus three. I rockets here. I am sorry. This is no longer here. Okay, so, uh, X plus one squared. Um, he&#x27;s X squared, plus two weeks less one. And if I look at these two expressions the next two expressions, you can not say that These are difference of squares, so X minus one weird and then Ah, sorry, minus three. I squared. So is his X squared. Plus two x plus one have been next. I have X squared when it&#x27;s two x plus one plus nine X squared plus two x plus one X squared, minus two x last thing say from outside these two expressions. You know X X squared. Sorry. Extra pounds for for seven x squared with 18 eggs less too. Okay, now this is key. Uh, pull the film you as asking.

Problem

[M] In a wind tunnel experiment, the force on a p…

Problem 33 Hard Difficulty

Find the interpolating polynomial $p(t)=a_{0}+a_{1} t+a_{2} t^{2}$ for the data $(1,12),(2,15),(3,16) .$ That is, find $a_{0}, a_{1},$ and $a_{2}$ such that
$a_{0}+a_{1}(1)+a_{2}(1)^{2}=12$
$a_{0}+a_{1}(2)+a_{2}(2)^{2}=15$
$a_{0}+a_{1}(3)+a_{2}(3)^{2}=16$

Answer

$7,6,-1$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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$7,6,-1$

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications