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Try It #4Find the zeros of f(c) 2r8 + br? Alx + 4...

Question

Try It #4Find the zeros of f(c) 2r8 + br? Alx + 4

Try It #4 Find the zeros of f(c) 2r8 + br? Alx + 4



Answers

Find the zeros of the function and state the multiplicities. $n(x)=x^{6}+4 x^{5}+4 x^{4}$

We were asked to find the zeroes off given function for fixed equal to X cube. Four times it is 24 x plus three X square here is to for it so we can express the given equation us. It is to forex into for X Cube plus three X Square equal to zero. But it is to forex. Cannot be equal to zero so he can write or excuse. Plus three x squared equal to zero or X squared in new for X plus three he going to zero. This gives us X can be equal to zero or executive to minus three by four. These are our solutions.

Given the function F of X equals X cubed minus seven, expert plus six X plus 20. You want to find all the zeros now you can list all the possible rational zeros which are positive and negative. +12 45, 10:20 Over one by the rational zero theorem and test until you get them or you can graph to at least find a rational one. And we see that five is a rational zero of dysfunction. And so we know five is a solution now to find the others. We synthetically divide five from the coefficient one negative 76 and 20 and we should expect the remainder zero dropped on the one. Multiply add multiply add multiply add. Sure enough. We do not what we have here we have expert minus two x minus four which we can attempt to a factor. Or we can use the quadratic formula. Mm This is your A B and C. In the quadratic formula we have opposite, be closer minus the square root of B squared minus for a C. All over two. A. Or to closer minus the square root of four plus 16 which is 20 over to that's two plus or minus the square root of 2020 is five times four. We can scrape up the four. That's two square root 5/2 which is equal to one plus or minus one. Spare route five. So one plus in one minus square root five. Sure. Yeah

So we have epidemics is given to us as X squared plus five x minus 14 and G of X. I'm going to write it underneath as X squared plus three X minus four. And we want to find the zeros of first Yeah. Plus G of X. And we know that means we want to add these two functions together. So let's happen. And that's why I'm lining them up underneath each other so we can add them. So we get two X squared plus eight x minus 18. And so if I pull out a common factor of two, yeah. Okay. Unfortunately that is not going to factor. Therefore why don't we just go back and use quadratic formula to find these solutions? So but I'm going to find quadratic formula and utilize this. I could also do completing the squares so we know that X will equal the opposite of B plus or minus the square root of B squared minus four times a time. See all over two times A. And I'm using just this part that I'm finding the zeros up and that's just gonna be too so I have negative four plus or minus and I have 16 minus four times negative nine. So that's gonna be 16 plus 36. And square root of 16 plus 36 is the square root of 52. And 52 is divisible by four. And this becomes four times 13. And this is all over to. So this becomes negative four plus or minus two square root of 13/2. And let's split that up like so so we get access equal to negative two plus or minus the square root of 13 of those are my two zeros for the sum. Okay And now let's find the difference part. Be asked us to find what F minus G of X has for zeros. This one is going to be much easier because when we subtract these two functions, the x squares cancel out. We subtract these two and we get to acts and we subtract these two and you gotta be careful negative 14 minus negative four. That becomes negative 10. And where does this equal zero? That's when two X is equal to tam and that's when X is equal to five. So that's where that function has a zero. Yeah. Now we want to find the zero of their product and let me come well we got some space right here. So we want to take F times G. Of X. And so we want to take those two functions multiplied together and find where the equal zero. And so the first function is that expert minus five. X minus 14, expert. I think it plus five X plus five x minus 14. And the next one, X squared plus three X minus four. Now we want a factor, we don't want to multiply that whole thing up and get 1/4 degree polynomial. That will be an enormous uh fourth degree polynomial that we're not going to know how to break down. And if I go seven and two and I have this one plus and that one minus, I multiply all this out. I will come back to here and here. I'm going to have X and X. And let's go four and one. We need a positive male terms. We need that to be positive for that to be negative one. And again, that gives me the right middle term when I multiply it out. And so now I can get my zeros. My zeros will take place when X is negative seven. When access to when access negative four and what access one. So I get four zeros because it would be 1/4 degree polynomial. So we had to deal with a quadratic situation, a linear situation and then 1/4 degree situation.

In the question we have to find the zeros for the effects which is X to depart four plus two, X cube plus eight X square plus 16. And where one of the zero sk is culture to Toyota now moving towards the solution, if K is close to two iota is one of the zero for the given fx, then K is close to minus two iota, which is the conjugate of Toyota will also Be a zero for the given FX. Now, as these boats are the Zeros, X -2 ι and ex bless to iota, both will be the factor of this fx. So now you have to divide the given fx by x minus two iota in two X plus two Y alta which can be written as fx divided by X square plus four. So doing this, you will get your questions as X squared plus two weeks minus two X plus four. So now your fx can be returned as fx is close to X square plus four in two x square plus two weeks plus four. Now we will be factory izing this term. Using the midterm splitting method, you will get x- Toyota into express to 1 ι.. This is characterized now this term will be X plus one plus route three iota in two, X plus one minus route three iota. From this, You will get access close to 2 ι access calls to minus two Y o. To access calls to minus one minus three iota. Access calls to one minus one plus route three iota as your roots. Thank you.


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