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0i 11 (10 complete)HW Scora: 63.6490Bkll BulldteQueslpn'Iflx)Help Me Solve TnisIroS:Question Help -lead ion usl Form polynomial f(x) with real coefficients ha...

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0i 11 (10 complete)HW Scora: 63.6490Bkll BulldteQueslpn'Iflx)Help Me Solve TnisIroS:Question Help -lead ion usl Form polynomial f(x) with real coefficients having the given degree and zeros_ Degree 4; zeros: 3+3i; multiplicity 2f(x) =alx - ( - S)llx - ( - S)llx - ( - 3+3 iJIlx - ( - 3 - 3 0)] Next we convert the factored form of f into the form ((x) =a ( bx? cx2 dx+k) where is any real number and and k are integer coefficients. Multiply the factors of flx) and combine Iike terms. Fill in t

0i 11 (10 complete) HW Scora: 63.6490 Bkll Bulldte Queslpn' Iflx) Help Me Solve Tnis IroS: Question Help - lead ion usl Form polynomial f(x) with real coefficients having the given degree and zeros_ Degree 4; zeros: 3+3i; multiplicity 2 f(x) =alx - ( - S)llx - ( - S)llx - ( - 3+3 iJIlx - ( - 3 - 3 0)] Next we convert the factored form of f into the form ((x) =a ( bx? cx2 dx+k) where is any real number and and k are integer coefficients. Multiply the factors of flx) and combine Iike terms. Fill in tne missing coeflicionts {(x) a[x - ( - S)Jlx - ( - 5)I6x - (- 3+3 ix - (-3-301 262 1Ox + 25) [* - (-3+3i)x- (-3-3i)*+(-3+31X-3-30] a(x 1Ox 25) ( 3x - 3ix+ 3x+3i*+ +9i-9i -912) 6 1Ox + 25) (2 6 * + 18 ) 1822 10x" + 607 180x + 2522 150x 450) a(x 6x8 a(x +-D-Ex+4so) in the edit fields and then click Check Answer: Enter your answer Clear All part Checlc Answver Close



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Find a polynomial function $f$ with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree $5 ;$ zeros: $1,$ multiplicity $3 ; 1+i$

Yeah. The concept in this problem is to take the zeros of a polynomial function and to write that polynomial function in order to do that. There are a few things that you will need to know to work with the problem. One thing is that if some number or expression is a zero of the polynomial function, for example, effects of one is a zero, then x minus x. M. One is a factor of that polynomial function. And then also the cons you get paired there, which says that if one complex number is zero. So is it common to get? So let's get started on this problem. We are told that there is a partner number of function of degree five. With these three seals given. Now, since this fall normal function has a degree of five. It can have at most five cereals were given three of them. But we'll be able to tell what the other two zeros are because of this kind get paired there. We know that if x my I'm sorry, we know that if x me back up one more time, we know that if negative I is a zero then its cons get I has to also be a zero. And if one plus I is a zero then it's contradict one minus I is also zero. Now with all of that in mind knowing these five zeroes, then we can write some factors. Okay, F two is a zero, then x minus two is a factor if negative, I is the zero than x minus a negative I is also see well I mean it's a factor of which I can write that is explicit and then also if I is a zero, an x minus I is a factor. If one plus I is a zero and x minus one plus a high is a factor which I can like that as x minus one and minus I. Mhm And if one minus I is a zero then I can say one. I'm sorry X minus one minus side is a factor. Which I can write that as x minus one plus I. So I have the five factors of my polynomial. So to write the polynomial. Yes, I'm going to say that this poem, I'm one of the sun real number. Eight times each one of those factors. So that X nine is chief. Yeah, times X plus high times X nine is uh huh. Trans x minus one minus I terms x minus one plus up. Okay, now, I know that a lot of hours will work to this. Okay, I'm gonna keep the X runs two for just a minute and I'm going to say that X plus I y times x minus I will be x minus x squared minus size square. Mhm. Okay. And then little trick here on this next part now it's both of those have x minus one. X minus one. So I'm gonna take that and go x minus one squared Minnesota scored. Yeah. Okay. Ah A little bit more out to work with this. Okay again some of the number eight times X minus two. This will become X squared plus one. Yeah and this will be X minus one squared plus one. Okay a bit more else will work. A lot of this is just gonna be multiplying these princes together. Big. Okay. The last princess. I'm gonna square this by now. We also have the X square manage to eggs plus one plus one. Mhm. Okay. Yeah it will be X square minus two. U. X. Plus two. Okay now a bit more multiply. Okay if I'm going to buy the first two parentheses together. The first two, that will become X to the third. That is two X squired. Mhm Yeah Plus six. Mhm. Yeah. Yeah. No it's too okay. I got that last process. Okay now it's a lot of multiply. I will start off and I'm going to take the heat to the third times everyone the terms of the second parenthesis. So I will have thanks to the fans managed to X to the fourth plus two eggs to the fired. Yeah. Okay now I'm going to take the negative two X squared terms everybody and I'm going to just kind of keep track of what all I have. I'm gonna wind up my like terms so I've got a negative two X to the fourth. It positive for extra the third. Yeah, negative forex choir. Yeah. Okay now I will take the X times each one of those and again lining up my like terms. I'm going to have extra thyroid man is two X squired plus two eggs. Okay and then lastly I will take the negative two times everybody get lining up negative two X squared plus four X. Name is four and I did like this. I haven't been lined up a little bit easier to collect my like terms. So now I can say that my function is some real number. Eight has that leading coefficient times X to the fifth minus four X to the fourth plus seven. Next to the third 98 K X squared plus 66 minus fault. Okay and that is the polynomial function whose zeros were given and who zeros were able to deduce from the can't you get paired there?

Yeah. In this problem, you're going to be given a couple of zeros of a polynomial function and asked to write to the polynomial function before we start the problem. They want to go over with a very important concept and that is that if x one is a zero of a polymer function, then x minus x m one is a factor of the function. Very important idea. Also a very important idea of this that we'll use in this problem is something called the conjugate pair there. And what this tells you is that one complex number is a solution or a zero of a polynomial function, then it's conjugated is also Okay, so with that in mind would start this problem, we are given the fact that there is a paul number function of degree forward. Yeah. And it has two zeros that are given to us of I and one plus two. I. And we are to arrive to the polynomial function with these conditions. Well, the contour get paired there and tells us that if I is a zero, opened its contour kit negative is also zero and if one plus two is zero. So is his college kid one minus two. So those are the four zeros of this polynomial function. And it can only have at most four zeros. Okay so now we're ready to try writing the polynomial function. So I'm gonna say that the function is some real number. A times ex Manisha times X minus and negative. I So I become explosive terms X minus one plus two huh? Times X minus. My mom is to uh now we need to take this and put it in a standard for So I'm going to do a couple different things on this step. I'm gonna go ahead and my father to princess of the front together so the text square of minus a square. And I'm going to do a little bit of rearranging, distributing the subtraction sign through the princess on the third and the fourth parenthesis. And then this is x minus one plus tour. All right. Now keep in mind the high square is equal to negative one. So any time we have a high square we can replace it with a negative one. Okay, so this first princess can be calm. Oops. His first Francis can become X square plus one. Now I'm gonna do something a little different with second Princess. I'm gonna write this as X minus one. Nine is too I was so why in just a minute and a similar type plan on the last parentheses? Yeah, yeah, if I continue working with that, I can take these two parentheses in, right that as x minus one, I'm squired Mattis for escort. Okay then I can take the last large parenthesis and I can take the x minus one and square it. That will be X square minus two. X plus we won and the negative four I square becomes a plus four. So now that parentheses is simplified to X square minus two X plus five. Okay so now it's just a matter of multiply and distributing so I got my a okay I'm going to start off and I'm gonna distribute the X square through that last parentheses. They'll give me extra. The fourth man is two X to the third plus five X square. Yeah kenya I'm going to distribute the positive one. Thank you that parenthesis. So they'll give me a positive x square mhm minus two X plus five collecting the like terms. Yeah. Yeah we end up with extra. The fourth managed to extra third plus six X square minus two X plus five. And that is our polynomial function. Now the value of A can vary, but this part in the princess will stay what it is because of what the zeros are.

The concept involved in this problem is to find the remaining zeros of a given polynomial function. We're going to be given some information about upon our function and we're going to be given some of the sea rose and we need to find the remaining secrets for this particular polynomial function. We're told that it has a degree of five. Yeah. Which means it has had most five cigarettes. Well given the zeros of zero, one, two and I and we need to find any remaining zeros, forgiven three zeros that are real numbers. But then were given a high avidity. There's a complex number. If you have a complex number that is a C zero Yeah. Of a function, then it's conjugated is also groups. This conjugation is also a zero. You must say the complex numbers as zeros come in pairs, the complex number. And also it's complex. It's cons kit So we have high as a zero then negative I is also a zero. And that will give us a five zeros of this particular polynomial function.

In this problem you're asked to Find a polynomial function to right of parliament a function of degree four. You seem to given Zeros. Now, since this polynomial function has a degree of four, we know that there is at most for zeros. Yeah, we are given that one of the Zeros is three With the multiplicity of two and then another zero it's a negative are if we know that 10 is a negative I The conjugated payer theorem tells us that the conduct of that is also a zero, so also positive I is a zero. And that makes up before zebras. Now, another important idea to keep in mind before you really get started to write in the function is that if let's say some X wand is a zero. Yeah. Of a polynomial function then x minus. Except wand is a factor for that polynomial function. Yeah. So if you know that three is a zero, Then that means X -3 is a factor. And since three is a zero with multiplicity of team, that means x -3 Squared is a factor. Okay. Fx If a negative I is a C rope, the x minus a negative I. Is a factor which on my right is X plus I. And then if I is a zero, the next minus a high is a factor of the polynomial function. So the poem, a function that we're going to write is going to be the product of its factors multiplied by some wheel number A. So we're going to multiply these factors together. Okay, so if you start off and square or by now milk, that'll be X square Matter six x plus nine. And I can multiply together the sum and the difference of the same two numbers. So that's X squared minus uh score here. Keeping in mind that I square is a negative one. So I can write this now as eight times X squared minus 66 plus nine times X plus one X squared plus one. Mhm. And I didn't multiply the two parentheses together. And I'm going to do this by taking the second princes and distribute it into the first. I'm gonna start off and distribute X square into the first parenthesis. Yeah. So that will give me extra before 9- 6. 6 to the third because nine X square. Then I will distribute the positive one into that. First proof the cease. That will give me a positive X square six X plus nine collecting the like terms super find this. I will end up with my polynomial function being some real No. eight terms extra fourth. Nine is six X to the third plus 10 X square. We have six x plus more. And that is the parliament function that has been given zeros along with the uh can you get that? We found


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