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Find all rational zeros and factor flx) f(x) = x 3 - 8x 2-x + 8 None of These -5,-3, 4; f(x) I(x 5)(x + 3)(x 4) 3, 5; f(x) = (x + 1)x 3)(x 5) 2, 8; f(x) [email protected] 1)x 8) ...

Question

Find all rational zeros and factor flx) f(x) = x 3 - 8x 2-x + 8 None of These -5,-3, 4; f(x) I(x 5)(x + 3)(x 4) 3, 5; f(x) = (x + 1)x 3)(x 5) 2, 8; f(x) [email protected] 1)x 8) 8; f(x) = (x 1)x )x 8)

Find all rational zeros and factor flx) f(x) = x 3 - 8x 2-x + 8 None of These -5,-3, 4; f(x) I(x 5)(x + 3)(x 4) 3, 5; f(x) = (x + 1)x 3)(x 5) 2, 8; f(x) [email protected] 1)x 8) 8; f(x) = (x 1)x )x 8)



Answers

Find all of the rational zeros for each function.
$$
f(x)=x^{3}+5 x^{2}+2 x-8
$$

Find all the rational zeros for the function. F of X equals X to the third plus five X squared plus two X minus eight. Well, what I need to do is factor this. And for a cube group, it's not the difference of two cubes of the sum of two cubes. What I'm gonna do is look at the leading coefficient is one and the last coefficient is negative. Eight So possible rational zeros. I take a look at factors of the youth. Let's just one for a We've got to. Actually, we've got one, 24 and eight. Those are all factors of eight. So I put each of these over one. So these are things I'm gonna try. Never use synthetic division and remember, positive or negative for each of these. So first of all, let me try one. And this is one X to the 3rd 5 extra the second two x to the first and negative eight. Remember, What I need to do is end up with zero right here. So I bring down the 11 times one is one and I add the five plus one is six six times. The one is 66 plus two is eight eight times. The one is eight negative. Eight plus eight is zero. Hey, so one is a zero for this function. And if I factor out X minus one, I'd be left with X squared plus six x plus eight that I could factor if it's fact herbal. So I have X minus one. And then this factors to be X and X Factors of eight to give me six are two and four, both positive cause they have all positives in here. So to get zero, either X minus one would equal zero or X plus two would equal zero or X plus four with equal zero. So X equals one which remember that was what I got here. That's why I knew X minus one was a factor. And then the 2nd 1 x would be negative, too. And the 3rd 1 Exit B negative for So these are the zeros of this rational equation the values that could plug in to give me zero for f of X

Problem. 19. It wants us to find all the rational zeros of this function f of X. So we can. So to begin, before we do anything, we noticed that we can facto an X from everything minus six. Next squares plus eight. And now we can really just because we have to find the rational zeros we can use the rash mall route Rational zero through, um, on this portion right here. And we also know that X equal zero is national zero because I hear this portion right here said it equal to zero boom. The solution. So but we can really focus on that red bit that I highlighted, and we can use the rational zeroth room. And that states that our which is irrational zero is equal to P over Cuba, where p by the values that could divide nicely into this constant. In this case, it's eight and cue are the values that could divide nicely into this coefficient of one that's that's didn't so then. Furthermore, our Q isn't equal to plus or minus one, and R P is equal to plus or minus one plus reminds, too. Let's remind us four and plus or minus eight. Awesome. Now we can write down our on our ours. Finally, it's just p divided like you. In this case, it's all the P values divided by plus or minus one. Well, when you divide by one is just gonna be the same thing that was in the numerator. So RPS this kind of we just rewrite them, as are ours, minus that looked really message plus plus Okay, it's plus miles. So now we can start kind of trying out some of these values and see what works. So just to, um, reiterate, we're working on his red bit here. We're not worrying about the action front so we can start by trying plus minus one. So, um, substitute a one in or we could a negative first. But whichever whatever floats your boat, plus a And for any way we can see that this is not going to, um, equal zero. So this does not equal zero. So we know that one is off the table. So let's try negative one really quickly. It's gonna put a negative year on negative here. This becomes positive one. This becomes positive. One times a minus six again that does not equal zero if you added up. So we know that plus and minus one are off the table. So let's try to now we'll substitute to in your minds to whichever one you want to try. First, we'll just try to and we see that not we see or substituting tearing for my six times two squared plus eight. And this kind of looks promising. We'll see. So two times or two to the four is two times. Two times two times two First is equal to 16 minus six times four, which is equal 2 24 plus eight. And what you look at that eight plus 16 24 minus 24. That's equals zero. There we go. So I'm gonna add that list so we don't forget it. So too, is also a national zero, and now we can continue trying the rest of these values. But I think it may be good to simplify our our expression that we would highlight and red because we found one of them. So we can just factor further. So using synthetic division on this red portion here, we can see what happens when you divide. Use of synthetic division. So we have a coefficient of one in front of the exit four and then we also have a zero for X to the three. We have a minus six for X squared. Then we have a zero for X, and we have an eight for the coefficient there. All right, so one plus here is 12 times one is 20 plus two is 22 times two is four plus four plus four by six. So we rave, saying that that's just equal to minus two. Um, two times minus two is equal to minus 40 plus minus. For it was minus for two times minus four is minus eight. And sure enough, it's equal to zero. And this simplifies out to I'm just going to write it down here with the little circle thing this is equal to So are ex stays the same. And that was read portion, uh, simplifies to x minus two times. Whatever was left here in this in the bottom part of the synthetic division, it's tough word that say so. We have X cubed. We have X cubed. Um, plus two x sward minus two x and then that's minus four. And then, I mean, we can just keep trying again, um, with these other best to see if we can maybe quickly find some other one where we can really quickly rewrite the rational Ruth Graham. Let's just do that. So now we're going to get, um, if we kind of fall along what we did up there. R P values in this case, so rational er is p over cute r P values would be the terms that can divide nicely into this negative four, and our Q values would be. It's still going to be the term that could divide nicely into one. So our Q values are going to be plus minus one, and P is going to be plus or minus one and plus or minus two and plus R minus four. So north side, we got rid of that ate up there, and now our values are just gonna be taking you to be ready for your hitting our peas because our Q values is plus or minus one. So then we get plus or minus one plus or minus to plus or minus four, so we have a bit um of more or be a bit less terms to try now. And we don't have to try plus or minus one again because we've already actually tried that. So we can kind of cross that off the list. Those are not them. And we've also tried positive to. So let's start trying. Negative two. And we're going to be factoring. This is written in terms of this red portion here, So let's substitute in a negative to and see what happens. So negative, too cubed plus two Think live two squared minus two times negative too. My as far. That's just equal to Well, negative two cubed well being negative. Eight plus two times. Well, negative two square does four times eight. Oh, I'm sorry. Times two, that's equal to eight. And then we get minus two times minus two as equal to plus four. And then we have a minus four. Cancel, Cancel. This cancels in that castle. So cynical to zeal and sure enough minds to work. So it's simplifying you. Sendek division on this red term here so so negative to here. And now we have a one for the X cubed. We have a two for X squared. We have a minus two for the acts of a minus four and then one plus zero is just one negative. Two times one is negative. Two two plus negative, too. Well, CEO zero times negative to that zero to negative. Two plus show is negative to negative. Two times negative two is four. Very good. Four plus negative for miss equals zero. And then now we can simplify this red portion of it further. I'm just gonna quickly right here so I can remember what it is when I scroll down the page. So what we had now is X Times X minus two and then we divided by X plus two. And oh, I don't even need to cooperate that I don't know why I did that. Then we have X squared plus zero x minus two. And here finally, or I forgot to add this to lift up here. Negative too. And finally we see that were left with this expression here and now, after maybe analyzing for a bit, you can come to the realization that the only answer to this is not going is not irrational zero. So we can suffice, Lee say that we have found all the rational zeros of this function as denoted by this box that this blue box

We have the function I have X equals nine X. To the fourth minus nine X cubed minus 58 X squared plus four X plus 24. We can use the rational zero test that says take the factors of our constant. Which there are a lot over the factors of our leading coefficient nine and we have all of our possibilities. Mhm. So let me start by. Oh. Mhm. My testing. This is just a lot. So what if we you can also check with a graphing calculator? If you were to graph, you would see all four solutions. I'll tell you three is a solution. And I'll just demonstrate using synthetic division how it could go further if you were testing. So we should confirm that zero is a remainder. Drop down the nine. Multiply add multiply add multiply and multiply adds, we get to zero. So then we have dyn x cubed. Well also from the graph, you would have noticed that another integer would be negative too. So, I'm going to now synthetically divide negative too from our coefficients at this point and demonstrate that we would again get a remainder of zero. Drop down the to multiply add, multiply add quantify AD. So now we're at nine X squared minus four equals zero, which we could just solve. Nine X squared equals four X squared equals 4/9. The square root of both sides, which is positive negative two thirds. Several positive are zeros would be um negative, two negative, two thirds positive, two thirds and three. Mhm.

We have given the full normal y equals two effects equals two x cubed, divided with X x group list it. Now we have to find the possible troops off. This pretty normal. So for this we can write it in the form off X cubed plus two que. So this can be done is X plus two and X squared less four minus two x So this is the first factor. And now for the second effective second factor weaken like the minus B plus minus B squared minus four A. C. By doing it, we get divided it to where they're doing it. So this will weigh by next four plus or minus 16 minus it. So this is not this will be square, so being is too. So this will be to plus or minus basically that this four minus for the subject of 16 they were headed with Do it so this will be one plus minus nor three. So they're the two factor X plus two x minus one plus little three I and X minus one minus three. So this is already quite but a normal inter effective. It's no


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