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Let To' T1. and 12 be real numbers with %o < 11 < *2. Write the Lagrange polynomials %o(2) , 01(2), and '2(2) associated with the nodes %o, 11, E2....

Question

Let To' T1. and 12 be real numbers with %o < 11 < *2. Write the Lagrange polynomials %o(2) , 01(2), and '2(2) associated with the nodes %o, 11, E2.

Let To' T1. and 12 be real numbers with %o < 11 < *2. Write the Lagrange polynomials %o(2) , 01(2), and '2(2) associated with the nodes %o, 11, E2.



Answers

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -1,1 (with multiplicity 2 and 1 ) and $(2, f(2))=(2,4)$

One we want to find it on, you know, meals off order 012 and three. So what is the definition? Hell if I know older end. So we just want to buy a crime. Crime Crime five times, right, So at this just need to the tracks. So, Prime Max, it's two times you duty to act Prime Prime Max. It's four times uh, the climax is just AIDS times. So when ACS is zero, the 1st 1 is just one. Peter always won by Peter is just after zero. It's just what on p one is first time plus two times axe, right? And he too, It's this time's passed the served time four times at square and, uh oh, divided by to electoral. Look at that on p three. It's one plus two acts us Tuapse square that same and US eight ready by three factorial access to the Sir. So this is for over three. Right? So here are the three terms

For the function. Awful bucks. You're going to want to overtax plus two. A funeral? Yes. He called you well off. First narrative Is the culture negative one over X plus to square I first due to about zero is equal to connective one over two square secondary. It'd be cool to to overtax us too. The power of three. I could look to your left at zero. Is he going to chewed over? She was cube, which is also you kowtow? No, Um no word to score. Third narrative. It's your call it too. Inactive three times three acts plus two to the power war. Furtive at zero is equal to negative. Two times three over shoe to the power of war. Which is it? What? Connective three over to to our three. Then we have the tail up. Anomalies about zero is my half. Other one is well, huh? Minus one over War comes axe after two is my half minus one were four acts. Plus I know what war that's next square over too. Are the three years Well, half minus one over four ax. Plus here this is I squared over eight minus. Read over. She was cute. Has X cube over three factorial. Which is it hot in here? This is in control. X cube over. Shoot to the heart of four.

Probably if I were to find Tyler. Well, you know, meals off orders 0 to 3. So here are the definitions. Thus, this formula here and when a is two and 1/2 axes. One the racks. A prime max is minus one, divided by the square prime prime lax. It's, uh, too. Or thanks. Kill a prime front. Prime axes six minus six. Divided by before I. So just you the formula here. Ah, Peter, relax. It's when axis equals toe to just have to. So it's one over two, right? And you are next is one or two plus some same times. Axe. What is this saying? I have Prime Max and when the access evolve too. So it's, uh, minus one force. Thanks. Oh, X minus two. Sorry, because when our ace you go to two, Here's axe minus a right. I forgot that he two acts issue goes toe. Just copy the puma An axe and plus a prime prime too. So that's one, of course, a cane and extruded too. Let's square and guided by selectorial. I don't forget that. So that's a nays. So the search term is one. The ray times X minus two square, and there's no use such terms here. And piece reacts is Peter Relax us, a minor sakes divided by so today four and Time's directorial, right? So it's, uh, no. Yes, it's us minus one or 16 and X minus two Q. Right. One over 60 minus. Here here's a piece reax.

Here in this exercise were given. Ah, zeros. Bigger one with the multiplicity, too. And they get two months. I won't have to find a phone little meal of minimum degree with real coefficients. Because we have real coefficients. We know we're gonna need another zero, and that is a complex con jacket of my complex hero. So I say I'm gonna be a polynomial in degree four because I have 40 stay. Let's go ahead and express it, in fact, it for him. First, we're gonna make it a function of X. So it's, uh, X minus my first sales next minus minus one, which is X plus one. I have most blessed Earth Resource to the third Power times extra miles. My 2nd 0 which is X plus two plus I times X minus one 3rd 0 for mine, for Ciro is Ah, what about that is, uh, multiplicity too. So it's squared, Not cute. And then x minus my $30 X plus two by the side. Okay, So these air complex consequence I know the eyes terms going to cancel 1st 2 problems way we multiplied all out. Demonstrated that. Now we're just gonna go ahead and say it's X squared. Um, plus two x plus two x plus, four x And then, uh, plus four minus. I squared is plus five again. The 1st 2 videos. 1st 2 exercises to foreign, possibly six. We went to every term because, but because their complex consequence huh? The eyes are always gonna kissel out. And, uh, eventually, you should be able to, uh, get to the point where you don't right all the terms and castle mouth, okay. And I'm gonna I'm off to my by no males one time. So x times x squared is x cubed plus four x cried plans for Iraq's and I'm moving out of my second term in my plus one times x squared plus X squared plus four x plus five. Okay, I've called out my scratch work. I'm coming back to a mathematical statement. I'm saying that this expression that this is just all right scratch work This expression here, x plus one times these two boys is equal to x plus one times extra Third, that's father X word student. What's next? Plus five? Yeah. Most lines here again by my foul. Actually the fourth. What's father X cubed. That's not X squared, plus five eggs. Also X cubed, my father explained. That's not an ex. That's fraud. Yeah, like about my terms. I get extra fourth but six x cubed plus 14. Next word. That's 14 X. It's a four tune plus five that isn't standard form, and it is a bone all rail of smallest degree with the three given zeros.


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