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Use the [cmainder theorem to find P(-2) for P() =-X 2x -5. the quotient &nd the Lemainder for the assoclated division and the value of P (-2) Specifically; give...

Question

Use the [cmainder theorem to find P(-2) for P() =-X 2x -5. the quotient &nd the Lemainder for the assoclated division and the value of P (-2) Specifically; giveQuotientRemainderP(-2) = [

Use the [cmainder theorem to find P(-2) for P() =-X 2x -5. the quotient &nd the Lemainder for the assoclated division and the value of P (-2) Specifically; give Quotient Remainder P(-2) = [



Answers

Use synthetic division and the Remainder Theorem to find $P(a)$.
$$P(x)=2 x^{3}-x^{2}+10 x+5 ; a=\frac{1}{2}$$

Here we have P of X, and we want to find the value of P of negative, too. And according to the remainder theorem, if we take P of X and we divided by X minus native to the remainder will be p of negative, too. So let's set up our synthetic division. We put negative two in the box and the coefficients of the polynomial go in a row. So we have one x to the fourth zero x cubed negative three X squared, negative two x and five. We have to put zero in place of the term that we're missing and the processes start by bringing down the first number multiplied by the number in the box. We get negative to write that in the next face and add the column. Take negative two multiplied by the number of the box. We get four right that in the next space and ad column, take one. Multiply that by the number in the box, write it in the next space and add the column. Take negative four multiplied by the number of the box, right that in the next face and ad column. And that's a Sfar as we can go. So that last number 13 is our remainder. So it's also the value of p of negative, too. He have negative two is 13.

So we're gonna be evaluating heave X for two separate values and negative two and five. So the first thing that we're gonna be doing is setting up our equation. So if P Vex is P and got negative two, we're going to substitute that negative too. And for all of the X values in the problem So we have negative two cubed plus five times negative too. Let's 12. So negative two cubed would be equal to negative eight on five times negative two will be negative. 10 plus 12 so we can combine some factors. So we got negative eight minus tens. We have negative 18 plus 12. And finally, we combine these two terms of negative 18 plus 12 will give us in negative six. So our first answer report A is when p is negative. Two equals negative six. So then we're gonna go ahead, and we're gonna do our second, um, problem, which is gonna be when P of X is five. So we have you've X equals five this time, so gamma gun and substitute this in, or each of the values of X, which we have here. So we're gonna have five cubed plus five times five plus 12. So, again, we're gonna go ahead and start simplifying each of these. So our first so five cubed is gonna be 125 plus five times five is 25 plus 12. So again, we can go ahead and combine. So 125 plus 25 is gonna be 1 50 lost 12 and that will equal 100 and 62. So that is our second answer for part B. So these are two answers. Yet he negative two equals negative. Six and P five equals 162.

Here we have p of X and we want to find PF two. And according to the remainder theorem, if we take P of X and we divided by X minus two, the remainder will be p of two. So let's put two in the box and get set up for synthetic division. We're going to put the coefficients of P of X in a row, so we have one x cubed five x squared, negative four x and minus six. Then the process is start by bringing down the one. Multiply it by the number in the box. So one times to write in the next space and add the column seven. Then multiply that by the number in the box seven times to That's 14. Write it in the next space and add the column. 10. Multiply that by the number in the box 10 times to that's 20. Write it in the next space and add the column. 14 can't go any further, so that's the remainder. So that's also the value of P of two p of two is 14

Aren't so. We have been asked to find p of a for the for this polynomial p vex by using synthetic division and remainder hero. So if we recall, I've written what remainder theorem is over here? It says that Pia, they is equal to the remainder. We get actually synthetically divide using song Given a just keep in mind that this division isn't just p of X. This will big, long polynomial divided bythe number negative too. That's not what we're dividing by. We're taking this big long polynomial, and we're dividing it. Bye X minus negative two or ex lost you, But for synthetic division, we don't need it to be written this way. But just keep in mind that were actually dividing by, uh, a Pellegrino view. So we're gonna start our synthetic division. We're gonna bring this negative two up in the corner, and we're gonna write all the coefficients of P of X. So we have an invisible one here, seven afford, but we're missing something. We have our third degree turn, our second degree term, our first degree term. But there's no number. All that means is that we're adding zero at the end that's our last coefficient. So we're gonna bring them all down 17 or and zero. And what we're gonna do is we're going to start our synthetic division. You know the first of this? We just bring that one straight down, and then every time we write a number down here, we remember we have to multiply by R A. So that's gonna go negative, too, because anything times one is just itself. Then every time I bring a number over to the next hole, I'm going to add that your numbers in that hole, so seven plus negative too is gonna give us five. Now that I've written a number down here, I know my next step is always just multiplied by a so five times. Negative two is gonna be negative. 10. I'm gonna add these two numbers that are in this thing called him again, and that's gonna give me negative six. And again, I've just written a number under the bar. So I know two multiplied by negative, too. That's gonna give me 12 on my last up. I just have to add these two numbers together, and that is going to give me 12 So if we remember with synthetic division thes I love you all of the numbers we've gotten except for the very last one These air the new coefficients on our second degree polynomial Because remember, we started with 1/3 degree polynomial and divided by a first degree polynomial. So that means we should end up with a second to reap all in one year old. This is supposed to be X to the power of one just a little bit hard to see. Yeah. So that's what these 1st 3 coefficients are. In this case. We would get X square, an invisible one in front of the X squared plus five X minus six. This last number is our remainder. This is our or so now that we have done the synthetic division and found our we know that p of A In our case, p of negative too is equal to our remainder, which in this case was 12


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