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

Question

In Exercises 17–32, divide using synthetic division. (2x^5−3x^4+x^3−x^2+2x−1)/(x+2)

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 the polynomial eight X to the five minus four X to the four plus two X cubed minus three X squared plus six X minus five. All divided by X plus one. Alright, so the first thing we want to do when we're dealing with synthetic division is just to double check that the divisor. So the denominator determine. The denominator is of degree one, which it is in this case. So we can go ahead and do our synthetic division. So let's just draw the table here. Okay, The first number we're gonna put it in our table, is that number on the left and that's going to correspond to the divisor or the denominator. Okay. And in this case it's going to be one minus one. Where does minus one come from? Well it comes from this X plus one term. What we're doing is essentially setting X plus one equal to zero to get X equals minus one. And then that's the number we use over here. So you're essentially looking for the root of the denominator. Alright, let's fill in the rest of this top row in our table. We're gonna use the coefficients from the numerator. So we're gonna start from the highest exponent work in descending order from left to right. We're gonna take the corresponding coefficient of each term. So the first one we have eight, the second term we have -4 which corresponds to the Um 4th order term Positive two corresponds to the cubic term, -3 corresponds to the quadratic term, Positive six corresponds to a linear term and then -5 corresponds to the constant term at the end. Okay, Alright, so let's go ahead and do the actual synthetic division process. So what we're gonna do is bring this eight down to begin with, Okay, we have a number at the bottom here. What's gonna happen to it is it's going to multiply to this -1, that's also outside the table. So we have eight times minus one, that's going to give us -8 and we're gonna put that in this space to the right and just above. Okay, now when we have numbers in the column like this, we're gonna take these and we're gonna add them together. So plus four minus eight is going to sorry minus four plus minus eight is going to give us minus 12. And then same thing, we're gonna take -12 from the bottom of the table. We're gonna multiply it with -1. That's going to give us positive 12. And then we have this column that's done, we're going to add it up. So two plus 12 is going to give us 14. Same thing 14 multiplied by -1. It's going to give us -14, we're going to put that in the space to the right and above, we can add up our column now minus three plus minus 14 is minus 17 -17. Again, we're multiplying to the -1 positive Adding up our column? That's gonna give us 23 And one last time 23 times -1 is -23. And then we're adding up our column minus five plus minus 23 is gonna give us minus 28. Alright so we've done the synthetic division now we just need to interpret the results. So the first thing I want to know is this final term here, this -28 is going to be the remainder. We can call it our and then to figure out how we write the rest of this. Well we have 1/5 degree polynomial, we're dividing it by a polynomial of degree one. So we know the resulting polynomial will be of degree four. Okay so we're just gonna start on the left eight X. And again the highest exponent is going to be four. So we start with four. Next we have -12 and again we're working in descending order. So we have four. Now we have exponents three. Next we'll have exponent two when we're using these values here to figure out our coefficients minus 17 X. And finally plus 23. And then in order to write a remainder, while our remainder is something that can't be divided nice and evenly. So we just have minus 28 over the original divisor X plus. And if you get confused and you don't remember what degree the polynomial should be. That you're writing here, you can always work from right to left. So you know that final term? The minus 20 is going to be the remainder. You can start on the right. Okay. So 23 you know as your constant term and you can work up minus 17 will be X. To the 1, 14 will be X squared and so on and so forth. Okay, so there we have it. This is the quotient like we wanted and that is going to match with solution D. Alright. Thanks for watching everyone see you in the next video.

In Exercises 17–32, divide using synthetic division. (2x^5−3x^4+x^3−x^2+2x−1)/(x+2)

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem