5

Given P(x), use the Remainder Theorem to find P(5).P(x) = x4 - 2x3 + 2x2 -4x...

Question

Given P(x), use the Remainder Theorem to find P(5).P(x) = x4 - 2x3 + 2x2 -4x

Given P(x), use the Remainder Theorem to find P(5). P(x) = x4 - 2x3 + 2x2 - 4x



Answers

For the given polynomial $P(x)$ and the given $c,$ use the remainder theorem to find $P(c)$. $P(x)=4 x^{3}+5 x^{2}-6 x-4 ;-2$

Half off acts do I did by eggs plus two. Then remainder is half off Negative too. Now we will use synthetic division to divide. So we write the coffee shakes off the poly Normal across the top. Full five maggot e cigs maga do full. And we are dividing by the factor X plus two No, on the outside this bar we put the solution which is negative. Two now here before comes down and four times minus two. So we get nag you to eight and we put in this here. Now we add them 45 Mnegative aid. So we get negative free negative. Three times negative to forget, Father do six. So we put here in the next Golam. Now we aired them, so he gets you. Now you can see that zero times minus two. So we get zero now. We had them negative. Full plus two. So we get now You do full So our finance or is half off Maggie to to equal to negative for that is over. Violence

All right. So her our question We've been asked to use synthetic division and arranger if you're, um, to help us find what p of a in our case, p of negative to ISS for this polynomial pew bets. So first, we're just going to use synthetic division to find, uh, our remainder, and then we'll talk about remainder here. So first we're gonna take p of X, and we're gonna divide it. Bye X minus a, which in this case, since AIDS negative too. We're gonna be dividing it by X plus shoot because X minus negative, too. Is that of the explosives to? So we're gonna use synthetic division to do that. I'm gonna take all of my coefficients. So I have my third degree term, second degree turn my first degree term. But I don't have my zero degree term, which is just my number. In this case, that means that number is just euro. So I'm gonna take all of this co visions this invisible one before the four and zero. And so we've done a few problems with synthetic division, so we know we bring down that first number every time I write a number down here, I'm automatically gonna multiply it by my A. So one times negative two is gonna be negative too. I write that in the next column and then once I do that, I add the two numbers that are in the same column that is gonna give me Chu. I've written a number down here again, so I'm going to multiply that by a which is negative too. So that's going to give me negative floor again. I have two numbers in the same columns will award you atom that's gonna give him zero. Now that I have zero down here, I'm gonna multiply that by my negative to appeared. That's going to give me zero again. And you know that zero plus zero is just zero. So in this case, what our new polynomial look like actually doesn't really matter to us, because all we're concerned about right now is the remainder. So in our case, our remainder is zero. So the reason that's important is because we're mangers. Nero tells us something kind of important. I'm gonna write our remainder zero. So we're arranging the room, tells us that for some polynomial p of axe when I to buy that by this linear factor X minus a where a is just so number. In our case, a was negative too. When we divide that we get so new polynomial que Becks plus some remainder. So in our case, our cue of X is going to be. We know that it has to be a second degree polynomial because we started with 1/3 degree and we divided by a first degree. So that news we're gonna start with an ex were we know that our first coefficient is one. So I don't have to write that invisible one our next coefficient issue. So that's plus two X and then I don't have to write look plus zero that comes after because anything plus zero is just itself. So in our situation, this is our cue of X, and there were, or this last number or remainder is zero. So remainder theorem told us that we take our polynomial. We divide By this linear factor, we get some new polling, a mule that's one degree lower, plus whatever. Remainder waas and remainder theorem says that all that p of a is equal to whatever that remainder waas so since our remainder was zero. We know that p of our A, which was negative too, has to be equal to our remainder, which is euro.

Our goal for this problem is to find P of three using synthetic division. So we put three in the box and the coefficients from the polynomial go in a row. We have two x cubed zero x squared one x and two Don't forget about the zero toe. Hold the place of the X squared term. Bring down the first number multiplied by the number in the box, Write it down and add the column and we have six. Multiply that by the number of the box. We get 18 write it down and add the column and we have 19. Multiply that by the number in the box 19 times three is 57. Write that down and add the column and we have 59 so the remainder is 59 that is also the function value. Soapy of three is 59

Our goal for this problem is to find P of two. So what we can do is synthetic division with two in the box and the coefficients of the polynomial in a row. We'll see what we get for the remainder, and that's going to be the function value. So we bring down the one, multiply it by the number in the box. We get to write it down and at the column we get negative, too. Multiply that by the number in the box. We get negative for Write it down and out of the column. We get negative. Three. Multiply that by the number in the box. We get negative. Six. Write it down and at the column, and we get negative eight. So the remainder is negative. Eight. And according to the remainder theorem, that is also the function value. Soapy of two is negative. Eight


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