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3.4.29Find an nth-degree polynomial function with real coefficients satisfying the given conditions. n=4; 2 i and 3 i are zeros; f( - 1)=100f(x) (Type an expression...

Question

3.4.29Find an nth-degree polynomial function with real coefficients satisfying the given conditions. n=4; 2 i and 3 i are zeros; f( - 1)=100f(x) (Type an expression using x as the variable. Simplify your answer )

3.4.29 Find an nth-degree polynomial function with real coefficients satisfying the given conditions. n=4; 2 i and 3 i are zeros; f( - 1)=100 f(x) (Type an expression using x as the variable. Simplify your answer )



Answers

Find a polynomial function $f(x)$ of degree 3 wilh real coefficients that satisfies the given conditions.
Zeros of $2,-3,$ and $5 ; f(3)=6$

This problem, asking us to find the third degree polynomial given these zeros and the fact that f of three equal six. So let's start with f of X so f of X, it's going to be equal to 12 Tuesday zero B X minus 23 Negative 30 bx plus three and +50 B X minus four and we would have some value a coefficient in front here. So let's go ahead and try and figure out what a equals. Using the fact that f of three equals six. So six would be equal to a and then anywhere there's an X we're gonna plug in three, three, plus three and three minus five. Let's simplify that. So six equals eight times 16 and negative two. That means that six equals That'll be negative 12 a viable size by negative 12 and you get a A is equal to negative one half. So let's go back to our original polynomial here, and we're gonna have to multiply this out so it be negative one half x x minus two X plus three X minus five. So let's do some multiplication here, so I'm gonna start with this binomial multiplication. So see what we get here. So we're going to have negative one half and that will be an X squared plus three x minus two x So I'll just be plus an X and then minus six. And this is being times by X minus five. So let's multiply that out now. And we can kind of distribute this X to each term and then this five to each term. So let's see what we get. We get negative one half and then it would be X cubed plus X squared minus six x and then it be minus five X squared, minus five X plus 30. Now let's simplify that. So we got some, like, terms here. So we again negative one half and we've got an X cubed. And then we've got an X squared minus five expose has to be minus four X squared minus four X squares. We've got negative six X minus five X minus 11 x and then plus 30. And this could be your completed answer right here. Now also might ask you just to distribute that negative one half in so you might get an answer of negative X squared over to mine. Plus, I'd be two x squared plus 11/2 X minus 15 if you distribute that negative one half in there.

The concept involved in this problem is to find a polynomial function given its zeros in this problem. We are given three zeros of a degree three polynomial function that were given these three cigarettes, We are also given the fact that if of three is equal to six. Now, if a polynomial function is of degree three then in factored form it can be written as F of x equals some leading coefficient A times the product of the three factors. Part three factors. So what we're going to do first of all is find the three factors and this is where the zeros will come in. If one of the zeros is too, that means that this possible function has a factor Of X -2. If another zero is negative three then there's a factor that's plus three. And with the third zero being a pause of five, then there is a factor of x minus factor. So with that information we can fill those factors in and say that f of x is equal to some leading coefficient. A Tom's X -2. Time's a factor X plus three. Time's A Factor X -5. Yeah. Now I need to figure out what is it? Lord leading coefficient. A And that's where knowing f of three. His six will come in when x is three F of x is six. So move substituting too this form and say six equals some leading coefficient a times three minus two times three plus three times three minus fact. So six will be cool Phative. What's negative? 12 times a making a a negative one half. Okay so with that information we can now go back And substituting negative 1/2 in for this a. So if if X. is equal to a negative 1/2. Tom's ex managed to 10 6 plus three times X minus back. And we need to expand this product. Mhm. So you would multiply X minus two times X plus three using your foil. And then multiply that answer terms X. Man is fat. And if you do that multiplication, you will then distribute the negative one half through the terms and get your final pollen. I will to be F. Of X equals a negative one half X. To the third. That's two x squared. Mhm. Plus 11/2 eggs. nine is 15. So this is the part on the function of degree three. That has the zeros of 2 93 and five. And f of three would equal a positive six.

We have a number 52 individuated to find polymer function effect and it we have given it zeroes. Zeros has been given toe minus three and five and also after people music. So let the poor normal function affects is a x minus two x plus three x minutes. Fire No, we have F three after he will be the value but plugging in X equal to 33 minus two three plus three minus five. We have after equal to six equal to a into one into six into minus two, so it will be equal to six by minus 12. That is minus one by two equal to minus on by two so portable function will be affects. Equal to minus one by two x minus two X plus three and x minus Fight. Thank you.

20 zeros are one minus 10 And the given condition is F two equals 23 Now we have to find the polynomial function of degree three. So by the number of zeros, two of them we have only fact owes eggs minus one, X minus minus one which is X plus one. An X zero which is X. Now the function of degree three is with real coefficient is F X equals to a multiplied by x minus one. Multiplied by X plus one, multiplied by X. Now to find the value of A. We have to use the given conviction that is F. Two is equal to three. So we can write F off two equals two. A multiplied by two minus one, multiplied by two plus one. Multiplied by two. Now have two streets. So we put three hair equals to a multiplied by one, multiplied by three multiplied by two. Now after simply five we get a is equal to half. Now put the value of A. In the given function. So we can write effects equals to have multiplied by x minus one, multiplied by X plus one, multiplied by X. Now to get the final, now to get the final polynomial function of degree three, we have to multiply this so we get after simplify half xq minus X, and this is our final answer.


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