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Pubd Arda third-Crgae (Uhat &, cubic) polynomial Q such tar Q(l) = -2.Q(l) = -4.0(1) = 6,and Q"(L) = 12 Me)Answersprogress)AnswerScoreReponucILbsu? Lalnuua...

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Pubd Arda third-Crgae (Uhat &, cubic) polynomial Q such tar Q(l) = -2.Q(l) = -4.0(1) = 6,and Q"(L) = 12 Me)Answersprogress)AnswerScoreReponucILbsu? Lalnuuatot

pubd Arda third-Crgae (Uhat &, cubic) polynomial Q such tar Q(l) = -2.Q(l) = -4.0(1) = 6,and Q"(L) = 12 Me) Answers progress) Answer Score ReponucILbsu? Lalnuuatot



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Find an equation of the cubic polynomial $f(x)=a x^{3}+b x^{2}+c x+d$ that passes through the given points. $$P(0,4), \quad Q(1,2), \quad R(-1,10), \quad S(2,-2)$$

In this problem, uh There is a given function expressed as f of X is equal to a X cubed. Just be X squared the C express D in a given condition says that this specific function passes through the following points P Q R N. S. So they know that if a given function passes through a given point, then we know that plugging in the X and y coordinates of that point to the function. We should we surely know that it will satisfy the equation. Now the end goal for this problem is to identify what the value of A, B, C, and D. So that we can find out what is the actual coefficients of the variables of the variable X. In this given function. Perfect. So for us to be able to do that, what what we first want to do is to plug in the X and Y coordinates of this point, speak U R N. S. The given function. So let's start with point P. So plugging in point T. To the function. We have zero comma fourth they know that the first coordinated ever since X. And the second coordinate represents the F. Of X. Or the Y. Okay so plugging in P. Accordingly to be on the given function will have four Is equal to eight times 0 Cube. Just feed them squared The seat and zero plus d. So both abc and yeah A. B and C. Are all gonna be zero because it's being multiplied to a zero value. That being said will end up having the is about for this one and this is the first variable we were able to calculate now for cues plugging in the values of Q. To the function will have two is equal to eight times 1. two Just be times one squared The see there's one. The steel We have two is equal to a. Just be the sea Last four. We know the value of for it we know the value of the it's equal to four. So we can plug in for um for the and this should give us a simplified equation and speed. Just be the C. Is equal to negative two. And it's called it equation number one. All right now plugging in point are which is negative one comma 10 will have 10 is equal to eight times negative one Cube. Let's be times negative point squared, Proceed as negative one with me Teresa. It's a bit messy. First D. Which is equal to pork. So we'll end up having six right here negative A. Because B minus Or let me just try to ride it this way. Let me release that one. Instead I negative a less B minus C. Is equal to six. And let's call it the equation number. Uh huh. I'm plugging in the last point point. Is It coordinates to -2 will have -2 is equal to eight times to kill just be banks. Two squared. See time's too this form The is equal to four. Okay. So we'll end up having excuse me. We end up having A. Does four B. The Stasi Sequencing to -6. Right? And let's call it equation number. So we have now three equations with three unknown variables A. B. C. This is now a system of linear equation and we have a bunch of method to solve this and I will try to solve this using elimination method. So the first thing I wanted to eliminate is variable C eliminate. See an equation one and two. And how are am I going to do that? So I will simply just add equation one. Nothing erased this one. I will simply add equation one with equation too. No that equation one is a plus b. The C is equal to negative and the question number two is negative. A. Just B minus C. C. Cold six. And they ignored by looking at it. It seems like if I will add this equation, I can actually eliminate two variables. So here let me reach this a little bit and let's say eliminate A. And C. in equation one and 2. So adding these two equations, we can cancel out A, cancel out, see and end up having to be is equal to positive four, Dividing both sides by two, will end up having B. is equal to. But this is the second variable we were able to calculate in this problem now we still need to find out what A. In C. R. So how are we going to do that? So what I am thinking is to um Maybe we can eliminate we can eliminate C. In equation one 10 3. And how am I going to do that? I will just multiply the equation number one by two. And then Sandra. Oh asian number two times equation one minus equation three. So let's try to do that two times equation one. Which is A plus B plus C equals negative tomb minus equation number three which is eight A. Does four B. Just to see is equal to -6. Now let's try to simplify the first equation. That should be equal to two a. That's to be just to see Equal to -4. Mine is the second equation equation number three 88 plus four B. To C. Equal to negative six. All right Drafting each term. So to -8 we have negative 68 2 4. We have negative to be and to c minus to see that canceled out. And negative four minus negative six. That should be positive too. Now from here, from the equation we were able to generate here. They thought that we can actually sold for a already why? Because we already know what the value of B. S And B is equal to two. So I'm plugging in the value of be here in this equation and that gives me the negative six A minus four is supposed to do Or -6 Day is equal to positive six, dividing both sides by -6 will have a sequel to Groups. Should be -1. This is not the value of pain. The only variable left and solve is variable C. And how am I going to soil for variable. See now you can pick any equation that we have written previously to solve for C. And let's say I wanted to use equation number one. So from equation number one at first A plus B plus C is about the negative to I know what is its negative one? I know what B is. It's positive to. So then I can solve for C. So this is one Plus C is equal to negative tool. So C. is simply equal to -3. All right. So for the question, we wanted to identify the actual coefficients of dysfunction and we were able to find out what A. B. C. And D. Here and um just to write the equation or dysfunction with its actual value. This should be equal to um Ever met equals a negative X. Q. These two plus two X squared See you think they read -3 x. And these four best four. So that should be the actual equation of this given function.

This is a question about systems of the linear equations in more than two variables. The question asks us to find the cubic equation whose graph passes through these four points. Mhm. In other words, we want to find which equation A. X. Cubed of be excoriated policy expose D satis phones or is satisfied. So sorry, satisfied by these ordered pairs. So we want to find the values of the variables A, B, C. And D. For which reason ordered pairs X. Y. That is fire this equation. Okay, so When Y is -6 X should be zero. Which is just deep. I'm sorry fo zero just gives us D. And so that will be a very helpful fact. As we go, just go about solving the variables. Next we have negative five Y equals negative 51 X. Is equal to negative one. Why is equal to negative 11 When acts as equal to one And why is equal to -14? one X. is equal to two. And so we have these four equations which form a system of four Equations in four unknowns Now. Well, four is the number. Um We already know the value of one of the variables deep. And so we really only have three unknowns and also three equations. Since this equation just tells us the value of D. These three equations are the ones that will help us find the three unknowns that are left. So why don't we just substitute in this value of D And solve the system of three equations and three unknown. We can put this in the form that we usually have With the constant terms on the right side by adding 6 to both sides of each equation. All right, okay. Now that we have that let's write a matrix for the system and will reduce the matrix. Yeah. First I want the studio zero And this to be a zero. And so I'm keeping the first row of the matrix. The same While adding the first row onto the 2nd room. And subtracting eight times the first row from the 3rd room. Yeah, Sorry, adding eight times the first room onto the 3rd room. Because we have a positive 18 and a negative one here. So I want eight plus negative age and then four plus eight, two plus negative eight and negative eight plus eight. Now I look at my matrix and notice that I've ended up with two rows. That can be simplified. That is there is a common factor of two here and six here. So I can now reduce those rules by those numbers. Okay, so this is the result of that here. This is the result of division of this row by two And this is the result of division of this row by six. Okay so next step, once I have these two zeros Is to solve their ticket. This to be a zero as well. Thanks. Can achieve that by subtracting twice the second room From the 3rd room. So subtracting this from the third row before I had divided by two. So maybe I shouldn't have divided by two after all. Okay, so since I'm subtracting twice the second room from the third room I have In the first entry 0 -0 0 in the second entry to Menace to The 3rd entry -2 Sorry -1 minus zero And the last entry zero -2 -2 -2s. And so I end up with this which if I then Well if I scale this third row by negative one I get that the value of the last variable is negative for. Okay so I have the values of these two variables. Now I just want the value of the last variable I can get by keeping on by still real reducing and getting these two entries to be zeroes. Yeah. So first of all this can become a zero if I subtract in the second room from the top room, the first row And I can get this to be a zero by adding The bottom row or the 3rd room. Yeah. And then not just tells me that the last variable is one Or the first variable is one. And going back to the original system we remember not to the first variable second variable and the third variable R, A, B and C. And so A B and C are equal to 1 -2 and a -4. And so the cubic function that we are looking for is a times X cubed as many times X squared plus C, times X. That's me. Remember D was -6, which we found right away. Yeah. Now to check our work, we can graph this and make sure that it passes through these four points.

In this problem we have given function express as F of X is equal to X cubed. Just be X squared the C express D. So this problem says that the following points P Q R. S process through this given function. So the end goal that we have here is to find out what the values of A B, C and D are. And they know before we start solving for this one, let's realize that if a certain black curve or function is passing through a given point, then we know that if we plug in the values the X and Y coordinates of that point to that given function it surely will satisfy. So now our first goal is plug in this point towards a given function. And in such a way that will end up creating a systems of linear equations with variables A, B, C and D. So let's start with pointy at zero comma negative six. So take notice full service. Your ex the first coordinate, The 2nd coordinate will be or why or your FFX. Okay so let me Teresa. So for substituting p point towards the given function will have -6 is equal to a. And zero. You The speed I'm zero squared the sea time zero. Presti So from here abc will just be zero because it's being multiplied by zero. So we'll end up having B is equal to negative six. So this is the first value we were able to soil for this point. I was substituting Point Q. On the given function. We have one negative 11. So -11 is equal to eight times 1 Cube. The Speed Times one Squared the C. Times one plus D. And so we'll end up having a josh B. The C plus D which is negative six Is equal to -11. I just switch the sides of this equation simplifying this one will have A plus B. Plus C. Is equal to negative 11 plus six. Touch me negative. Right? And that's called it equation number one, plugging in the values for points or Which is -1 -5. We'll end up having negative five is equal to eight times negative one q. That's three times negative one square. Let's see times. Hold on. Plus C. Times native one plus D. I'm simplifying this one further. We have negative five is equal to negative aid. Press B. Find A. C. S. D. Which is negative six. So we can simplify our equation as negative A plus B minus C. And then we combine negative six to the other side of the equation that becomes negative five plus six. So this should be equal to positive one. And let's call it equation number two. And lastly, let's plug in point S We coordinates to negative 14. She Would have negative 14 People to eight times too cute. three times 2 squared The c. times to just be. So we'll have negative 14 sequel to eight A. That's four B leads to C plus negative six. Finally we'll have eight a. Trust for B plus to see equal to negatively. Right now the 14 plus six is negative eight. And let's call this one as our equation number three. And actually before before we proceed to the next step, I just realized that this equation right here, It has a common multiple. Everything can still be divided by two. And that being said, I will divide every each of every term by two. So we'll end up having four A. For us to be that she sequel the negative for. And this one is what we will call equation number Okay, now that we have three equations with three unknown variables A. B and C. We can apply a bunch of methods elimination, substitution or matrices to solve for these variables. And let me try to use elimination. So let's try first eliminate um we can eliminate um A and C. An equation number one and two in one. And By simply adding up this equation one and 2, I will be able to dominate A. And C. So we have E. The B. The C equals a negative five for equation one and we have negative A plus B Minour C equals to one for equation number two. And what we're going to do here is to at both of them and A plus negative A cancelled out C plus negative. See canceled out. So we end up having to be is equal to negative for Dividing both sides by two will end up having B. is equal to -2. Okay But we still need to find out what um A. N. C. R. So what I wanted to do here is um to plug in B. two, equation 1 in three. If I will do that I will simply have a plus. For equation number one will have a plus negative two. The C. Is equal to negative fight. I'm actually using equation number one but this time I'm plugging in the value of B. So this will give me a policy is equal to negative three. Negative five plus two is simplicity. And I wanted to all this one equation number or and now we want to plug in this as well. We want to plug in B. As well to equation number three. So for equation number two we have 48 just to B plus C. Support a negative force. I know what he is. It's negative tune. So that Gives me an equivalent equation of four minus four. Let's see. Support the -4. or simply for E. The sea equal to zero. And let me call it actually it's nicer to express. See in terms of a Let me call it equation # five. Okay. And lastly, what I wanted to do is the substitute equation fried equation number four. By doing that I should be able to solve for the value of the we have A plus C. Is equal to negative three from equation for but equation five says C. Is equal to negative for A. So we have a. Plus the value of C which is negative for A equals a negative thing. And we'll have here negative three A. Is equal to negative three. Then we know that A. Is equal to one by dividing both sides of the equation by negativity key. So we know a you know, be we know the we can also foresee using equation number five. So from equation number five, this 1 18 We say C is equal to -40. That being said sisi was negative four times one, or C is simply equal to negative form. We now know the values of A, B, C and D. And so we can rename this function right here. So f of X is simply equal to a. Which is one. So XQ The value of B is -2. So this should be minus minus two X squared and C is negative four minus four X. And the is -6 -6. So this is the actual coefficients of this given functions.

So here we are given a quarter Immigration That is PX- 2ared plus Q X plus are Is equal to zero. And it is given that the ratio of the root of the roots of the squatter equation. He's three years to four. That is if alpha and beta are the roots of the squad of the equation, then alpha is to be to is equal to three is too four. So we need to find out the condition with such satisfies the given equation. Okay, so let us consider the common common term be right. That is alpha by beta is equal to three way by forward. Therefore alpha is three ways and peter is equal to four way. Okay. No from the court litigation we consider this product and consider the product and some of the routes that is alpha plus. Better physical too three y plus four way That is seven by is equal to negative B. Bay that is negative cube I. P. Similarly product that is alphabet A. Is equal to three way in the four way, which is equal to oh 12 Y squared is equal to R. I. P. That is why score can be written Ernest Are by 12 p. And similarly why can be written us negative goodbye seven people. So we have the value of why. Therefore. So we substitute the value of Why in this equation that is Negative Cube I. seven p. The whole square is equal to Are by 12 people. That is Q. score by 49 p. is equal to are by 12. So we can tell PNP so we get the conditioners 12 years ago, 49 hearts. So this is the condition which satisfies to give an equation.


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