In Exercises 17–32, divide using synthetic division. (x^2−5x−5x^3...

Question

In Exercises 17–32, divide using synthetic division. (x^2−5x−5x^3+x^4)÷(5+x)

Accepted Answer

Hey everyone! Welcome back in this video. We're gonna be using synthetic division to find the quotient of the following polynomial. So we have eight X squared minus four X minus two X cubed plus three X to the four. All divided by two plus X. We have synthetic division. The first thing we want to do is just double check that the divisor or that term in the denominator is the degree one polynomial. So in this case we have two plus X as the divisor. The highest exponent is one. And so that is going to be a degree one polynomial. So we're okay to go ahead. Now the numerator we're given is eight X squared minus four X minus two X cubed plus three X. To the four. Okay. And what we want to do with that is we're just going to rewrite it. Okay, when we do synthetic division it's important that when we write out our table the coefficients go from left to right, decreasing from highest exponent to lowest exponents. So what we want to do is we just want to rewrite this from highest exponent to lowest exponents. So that it's easier to convert it into our table. So the highest exponent we have is X. Before. So we're gonna write that term, first three X to the four -2 x cubed Plus eight x squared minus four X. And we don't want to forget about a constant term. So we'll add in the plus zero here just so that we remember that there is indeed a constant term. Alright. So now we can get started with our table as we draw the table that we use for synthetic division. And the first number we want to fill in is the one to the left of the table and in this case it's going to be minus two. Okay, now where does that come from? Well it's going to come from the denominator or the divisor that we talked about earlier. Two plus X. So in this case we essentially set two plus X equal to zero. Then we solve for X to get X is equal to minus two. And that's the value we use in our table. So it's like finding the root of that denominator. Alright, so to fill out the rest of that top row in our table, we're gonna use the coefficients from the numerator from that polynomial that we rewrote. We're going to put the coefficients corresponding the highest exponent on the left. All the way down to that constant, zero that we added in. Alright, so we have three corresponding to X. The four minus two corresponding to X cubed eight corresponding to x squared minus four corresponding to the X term and finally zero are constant term. Alright, so to start our synthetic division, we take this three here on the left, we bring it down underneath the table and we leave it. Alright, so this term on the bottom is going to multiply by the -2 on the left. So three times -2 is going to be -6. Now we're going to add within the column. So we have minus to plus -6 that's going to give us -8 and we're going to write that down below. Alright so now we have -8 and again this number at the bottom is going to multiply to the -2 on the left -8 times -2 is going to give us 16. And again we're gonna add within the column And write it below 24. Right now we have 24 in the bottom, 24 times the -2 on the left. That's going to give us -48, adding in the column minus four plus minus 48 is going to be minus 52. Okay. And one final time the minus 52 on the bottom times the minus two on the left hand side. That gives us 104 and adding that final column, zero plus 104. We get 104. So now we've done our synthetic division and all we need to do is interpret the result. Okay, so the first thing we want to do this right hand number of this last number on the right 104 in this case this is going to be our remainder. We can call it are. Okay. All right now looking at the solution, we started with a polynomial whose highest degree was four. The four here that we wrote out. And we're dividing it by a polynomial of degree one. We already talked about that divisor being of degree one. So we're dividing by a polynomial of degree one. We're going to reduce that four down by one. So our new function is going to start with an execute the highest exponent is going to be X cubed. Alright. And the coefficients, these numbers that we found in our table are going to correspond to the coefficients. So starting from the highest exponent X cubed. And starting with our first number in our table, three is going to be the coefficient. Okay, next we have a -8 in our table and we're working down in our exponents. So that's gonna correspond to X squared. Then we have the 24 in our table And that's corresponding to the X to the one term or the linear term. And finally we have -50 to that value from our table corresponds to the constant term. The last thing we need to add here is a remainder. So we have 104 and the remainder is going to be still divided by the original divisor. Okay, It couldn't be divided Nice and evenly. So that's the leftover. So divided by two plus X. So here we have the solution three X cubed minus eight X squared plus 24 X minus 50 to plus 104. All divided by two plus X. And that's going to correspond with answer C. All right. Thanks everyone for watching. I hope that helped see you in the next video.

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In Exercises 17–32, divide using synthetic division. (x^2−5x−5x^3+x^4)÷(5+x)

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