In Exercises 49–50, find all the zeros of each polynomial functio...

Question

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. g(x) = x^4 - 6x^3 + x^2 + 24x + 16

Accepted Answer

Hi there. This problem says you want to express his polynomial as a project. Was later in quadratic factors. We first define all zeros of this polynomial polynomial is X to the fourth minus two. X minus three squared plus four X plus four. So we have to make use of what's called irrational 00. So you should remember this from your lesson. But rational zero theorem states that if we have a polynomial where P is the coefficient of our final number which is a zero, which would be a very less number. And we have a Q. With the coefficient of our very first number or it's on the higher degree term. So okay, it's again then we have rational zeros such that plus or minus factors of P over factors of cute. So what I mean by that as we have P equals zero. A zero is our very last term. So if you look at our numbers here, you have a four at the end of a zero should be four. Similarly for our Q. It is the coefficient on the first degree that we have in our equation which is X to the fourth our highest degree. So that is a four. Which is one coefficient from the X. The fourth is one. So now we can write out our factors are factors of four. We're gonna ride them out or we have one in four and two and 2 Are factors of one are just one and 1. So we're calling her right out all of our possible fractions. Yeah. Plus or minus Batches of p. So we have 4/1 Plus or -2-1 and finally plus or minus 1/1. So our factors will be plus or -4 Plus or -2 and plus or -1. These are possible answers for zeros. Now see if we can figure out which one of these are actually the zeros do that. I'm gonna take every one of these numbers and plug it into our equation. We do that. We should be able to see if we have any of those equaling zero which would make it a zero. So let's see let's go ahead and do this first. We have exit 4th As to X 3rd minus three X squared plus four X plus four. Let's go ahead and let X equal our first one positive form. 4th to the 4th -2 times 4th 3rd minus three times four squared plus four times four plus four. It's going to simplify this. We get -1. minus 48 Plus 16 Plus four. Go and add all that together. That is equal to 100. So that is not a zero. Let's try negative four. Just to check So you have negative fourth to the fourth minus two times negative fourth. The third minus three times negative four squared Plus four times 84 Plus four. That is Plus 1. 28 minus -16 plus four which is also a very large number, 324. That is also not a zero. So just keep checking all these until we figure out which ones or zeros. So we'll keep going here equals two to the fourth minus two times two to the third minus three times two squared plus four times two plus four. Which we simplify that. That is 16 minus minus 12 plus eight plus four. What should we check this? We have zero minus 12 is 12 plus 12. We have zero, so X equals two is one of our zeros. Well, keep shaking all of them, X equals negative two. You get negative two to the fourth -2 types of data to 3rd minus three times negative two squared plus four times negative two plus four. Check that -16 plus -12 -8 Plus four. Which simplified that is a native 16 which is not one of them, X equals negative one. Actually X is positive one, next to the fourth minus two times one to the third minus three times one squared plus four times one plus four. That is going to be one minus two minus three plus four plus four. So that turns out to be one plus five or +45 plus four is nine -5, that is four, that is not zero. Finally, let's check our final zero possible zero negative one, negative one to the fourth minus two times negative one to the third minus three times negative one squared plus four times negative one plus four which is positive one plus two minus three minus four plus four. And all those together we get three minus 30 minus four plus 40. So we have zero. So we have our two zeros. Now X equals two and X equals negative one. So we can look at the answers and we see what our answer is. Let's figure out how we would figure out the exact answer to find that. We would use synthetic division. So state division allows us to divide our numbers to see which zeros are which And if it's an actual zero or not are zeros we're gonna go ahead and attorneys and the factors or expressions Someone subtract two on both sides. two and there's two. Besides I get X - and over here we would have the same but X plus one. Now it's going to go back to our original equation. X to the 4th -2 X to the 3rd as three X squared Plus four x plus four. Let me synthetic division and divide one of our factors. Let's say we use X plus one first. So I'm gonna go ahead and say these are coefficients in our city division. We put our coefficients of our numerator at the top. Our numerator equation is going to be our long equation are X to the fourth equation. So we have one -2 -3. 4 and four. Now on the outside. The outside will take our zero X. Equals negative one. So I'll put our negative one there. So our first thing doing state division is carried down our leading symbol carried on that. We get what now our next thing you do is our next column. We're gonna take our number out here, multiply it by this number down here and put it into the right below negative two negative one times one is negative one. Now we're gonna add those numbers together so negative 2 -1 gives us a negative three. Now we go and do the same over and over until we get to the end of the polynomial. But yeah -1 times negative three is 3. We add them three times three plus three is zero. Don't do it again For state of one. I'm saying 81. 0 and zero bring down to four. Finally we get four times negative one is negative four. We get zero. Okay so now we have a remainder of zero at the end which is good. That's what you want. So we know that this is X. The fourth. Originally when we divide out one of our numbers, this turns into X. The third. This turns X to the second extra one and that's our number at the end. So we have now X plus one multiplied by X. The third -3 X. II. Um zero X to the one and plus four. Let's go and check one more time. So we're gonna use this synthetic division one more time. We have X the third which is one X three X the first zero. And finally we have four at the end. And let's say we have negative one again, bring down the 18 to 1. Times one is negative one. And we get a native three minus the 911. Or plus the native one is native four -1 times four is 4 -4. Zero plus four is 44 times eight of one state of 40. We once again have the remainder. So go and do that again. You have an X squared here X to one and just coefficients left. X squared minus four X plus four. So if we go back over here we have X plus one squared times X squared minus four. X plus four. Okay, now we can confirm again by dividing once more if you wanted to with that division to determine if there's another factor, another multiplier for X plus one. But you can also notice here that this X squared minus four X plus four factors out two x plus one squared. We can factor this, see any method but we also know our other one was X equals two. So we should know that this should be x minus two squared If we have to factor that out. So you know, the answer should be X plus one squared times X minus two squared. That if you look at our answers, our answer should be answer number C. Or let her see. Alright. I hope to help you solve the problem. Thank you for watching. Goodbye.

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