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Determine the equation; in factored form, of f (x) Then determine the remainder when divided by (x + 1)...

Question

Determine the equation; in factored form, of f (x) Then determine the remainder when divided by (x + 1)

Determine the equation; in factored form, of f (x) Then determine the remainder when divided by (x + 1)



Answers

Use the Remainder Theorem to find the remainder when $f(x)$ is divided by $x-c .$ Then use the Factor Theorem to determine whether $x-c$ is a factor of $f(x)$ $$ f(x)=2 x^{4}-x^{3}+2 x-1 ; x-\frac{1}{2} $$

All right, So we have another remainder. The're, um um so we're trying to find the remainder question Mark, Um, maybe I should explain this a little different than previous videos is Ah, you want the opposite of this value. So right here, I want the opposite. So I'm need to plug in X equals negative one. And for all of these excess and what the remainder theorem tells you is the answer for this, Uh, when I plug may have one into this will equal the remainder. As if I divided by X plus one eso as I do this negative onto an odd exponents stays negative and want any power stolen naval into and even powers positive and then minus woman plus one, you'll actually notice. And negative one plus 10 minus one plus 10 is your final answer. So what that tells us, if you're curious, is that if you were to divide by X plus one, uh, I guess X plus one is a factor of this problem.

We have the value off for fix are for fix That is equals true. Do you actually part or minus? Sorry it's bless X cubed minus three x less when we divided by X plus one by three Now we need to find the value of remainder as the last. We also need to take whether this given Sorry, but experts one but one by three Factor off our for facts or not. Okay, So in order to find the remainder, we are going to place expert minus often my three so are four minus on the street that the acquittal three might applied minus one by three over the 54 bless off minus one by three, won't you? Minus three and two minus one by three less when an engine 12 we will have Since we all know that minus one by three words are all that is equals to one my free will depart for So it's like one by three gold But so even made written part for three millions not to depart for my so it didn't come. Thank you. Ok Similarly will have now minus one by three. You against these two. Don't feel cancel out each other similarly will have no three since negative nearly exactly become Porter. Three My plane on mysteries of the board with each other less when so now we have one plus one equals two, which is definitely not equal to zero. So now we can say that our X plus one by three that is normal. Factor off fourth ex by Yuri Factor to him as factor totems. And if FFC that ID equals two geo them explain to see factor off FX and X minus. See the battle for fixed, then off the that is equal do Geo. This problem is definitely not falling the conditions for factor toe So we can say that X plus one by reason of off. Thank you.

We have the value off for fax that is equals to minus for X Q. Bless five X squared less, which is divisible by X unless three. Okay, so basically, we need to find the remainder as the last. We have project that X Plus three is a factor, or for X or no. Okay, so now what we are going to do, we have expressed three. So it would like we're going to the place except minus three. So, like minus four, it's minus four minus three. Thank you. Bless. Fight. My deflated. Minus three. Old square, less off it. And then controlling. We have minus four. Might reply with minus three whole groups of that week with minus going to seven. Bless. Fight Might reply with minus three old square that is equal to nine. Bless off it. Certainly minus four might be played with minus 1.7. So this were these orders will become 40. So full in 27 but 5 to 9. So forth. Multiply victory seven plus Find medical with nine plus off. Eight on soiling that B will still 1 61 feet. Obviously not able to do this, we can say that X Plus three is Northerner Fechtel four. Fix by Yuri Factor two. As active in states that if X minus c is a factor off affects, then 50 that will be used to G and Hinton using that term, it is clear to us that wants to when his body was to veto. Therefore, expect three. That will be that, nor be a factor focus.

So the question here, um, asked us about the remainder theorem, and it wants us to use that particular, um theorem in order to determine whether X plus four here is going to be a factor of f of X here. So if we recall here, um, the remainder theorem points out that if you divide our polynomial f of X by a factor of X plus four here, um, then you will get a zero remainder here. So, for example, here, uh, let's say if f of X is, um I mean X plus four here is indeed a factor of f of X and the remainder after the vision by that particular thing is going to be zero. So we can write this as f of X is equal to x plus four, multiply by two x six plus 1 29 execute plus 64 here. And what we get is that by synthetic division, this will also equal to zero here. So, um, what is the factor theorem? The factor theorem is just the reverse of this particular remainder theorem. If we eat synthetic, divide a pollen mobile by X plus four and get a zero remainder, then it's not only, um, not only is X plus four here or X equals negative four, so X equals negative for a zero of the polynomial, but X minus four is also going to be, um, a factor of this particular polynomial here. So, um, in this particular case, if we do the division, we get that, Yes. This is going to be a factor of FX.


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