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What condition should the coefficients of a polynomial satsify to ensure that $x+1$ is a factor of that polynomial?...

Question

What condition should the coefficients of a polynomial satsify to ensure that $x+1$ is a factor of that polynomial?

What condition should the coefficients of a polynomial satsify to ensure that $x+1$ is a factor of that polynomial?



Answers

What is a polynomial in $x ?$

Okay, so we want to find all of the factors of this polynomial X cubed minus three x plus Two were given that one of the factors. Zac's minus one. So we want a first use synthetic division to find out what comes next. But to know what to play again, we have to send our factor of X minus one equal to zero, which gives us X equals one. So it's a one that we plug into the synthetic division. Now we have to take our coefficients and put them across here. So the 1st 1 is one because it's one x cubed. But right here there is no X squared term. So we have zero x squares. Then we have our minus three X and then our two. Because there's no X squared term. We have to put in that zero. Otherwise we will not get the craft answer. So in our synthetic division, that one drops one times one is one zero plus one is one one times one is one negative. Three plus one is negative too. They got two times one is negative too. So we know for sure that that was a factor using these as our new coefficients. We have X squared. No sex minus two as a new term here they knew pollen over. Well, this is just a simple quadratic equation, and we already know how to factor those. You can use just about any method you want, but here I know that one negative too. Or one and two are what multiplies to give me a negative too. But does the negative go with the one or the two? Well, because this is positive. We know we have to have negative one and the positive to give us a positive X turn. So if we bring this down right here as an X minus one, that was our first term. Well, they end up with as our final solution X minus one squared times X plus two

So we're given the X minus. One is a factor of this polynomial executed plus two X squared minus X minus two. We're test with finding the other two. So first we have to find our divisor. To do that, we set our factor equal to zero, but with one to the other side. So we know our device here is well, now we can set up our synthetic division. Coefficients are one two negative one negative too. Dropped down the one multiply by 12 plus one is three one times three is three negative on plus three is two one times two is two And there we go our radio zero confirming that we did the factor correctly And this leaves us with quadratic equation of X squared Lost three x plus two We know many, many ways to factor this, but one and two are one multiply to give us too and wanted to add to three. That means our remaining factors must be X plus one and X plus two

So we're giving this polynomial and we want to find the remaining factors given the X Plus One is a factor. So let's use synthetic division. But first we have to determine our divisor, which, when we subtract one from each side, gives this X equals negative or so. Let's use negative one. Our coefficients are one exit fourth to execute two X squared negative two acts and negative three. So, after dropping down, the one multiplied by negative one gives this negative one. Troopers negative one. It's one negative. One times one negative. One two plus negative 11 negative. One time is negative. One. He's naked One day you two plus negative one is naked of three negative, three times thinking one. Is it positive? Three. So there we go. We're left with X cube plus X squared plus X minus three. Well, if you were to graph it, you get this little thing with this hook, then it goes up like that. This is that a positive one? So let's try X minus one, as are other factor. Or here's one with our 11 one negative. Three dropped down the one one times one is one one plus one is two. One times two is two one plus 23 one times three is three and well after this. So what we're left with now is X squared, plus two acts plus three. But that is not something that actually factors, and that's because it's going to end up with an imaginary result. If you set up buying the discriminative you're discriminate is B squared or four minus four times a time. See, which is four times one times three or 12. So this has imaginary results, so it's not a factor. So that means are factors where X plus one, X minus one and we can leave The last part alone has X squared, plus two acts bust three.

Okay, So to find we're gonna find a polynomial with zeros of X. Is it good toe one plus I and the zero of one minus, I we could put X is in front of these Caesar are zeros, and we're gonna work backwards in order from the polynomial. So we are going to subtract the right from the left in order to get back to zero. So x minus one minus. I on this side is equal to zero. These are zeros. And on this side of X minus one plus, I is equal to zero. So working backwards again. And now we're going to multiply these terms minus one minus I X minus one. First I Okay, now we're going to go through each term and multiply across that we're gonna do X times X X minus One times X minus one and x times I They were gonna do that with all the terms. And when we do that, we get a long turn. We have X squared. Plus I x minus X minus X minus. I plus one minus. I X minus. I squared. Plus I Okay, so now we have this long thing, But what we can do is we can go through and cancel things out. So for instance Okay, X squared stays here. We have an I X plus I ax. And here we were minus I X that we can cancel these out. Okay, here we have a minus. I in a plus, I and OK, so there's air. That's everything we can cancel here minus X minus X is equal to negative two X. Okay, We have a one here. And then we have negative, I swear, aired. So negative I square. We know that I is equal to the square root of negative one. So if we square that we just get negative one negative negative one. It's just positive one to be up that end of getting the polynomial of X squared minus two x plus two and that, well, give us the zeros up here and it has a degree of to This is our final polynomial and feel free to check your work


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