In Exercises 17–32, divide using synthetic division. (x^7+x^5−10x...

Question

In Exercises 17–32, divide using synthetic division. (x^7+x^5−10x^3+12)/(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 polynomial were given our two X to the seven plus three X to the five minus five X cubed plus four divided by X plus one. Now, synthetic division. We want to first double check that our divisor or the term in the denominator is a degree one in this case. It is. So we can go ahead with their synthetic division. So the first thing I'm gonna do is I'm gonna take this polynomial that were given the numerator and I'm just going to expand it. Okay so are synthetic division needs to account for every single term? Um Every single degree of exponents even if they're missing. So we want to write it out like X two x minus seven plus three X five minus five X cubed plus four is what we're given. And we're just gonna write the missing terms. So two X to the seven plus zero X. To the six plus zero plus three X to the five. Sorry plus zero X to the four minus five X cubed plus zero X squared plus zero X plus four. Okay so now we have the coefficient for every term from the highest exponent. All the way down to the constant term. Okay so let's go ahead and right out our table for synthetic division and the first value we're gonna put in here is this term on the left. This is going to correspond to the divisor or the denominator. Okay and that corresponds to this X plus one. And from that we essentially can set it equal to zero, X plus one equals to zero which tells us X is equal to minus one. Okay? And that's the value that we're putting in here. It's similar to finding the root of the denominator. Alright next we want to put the coefficients of the numerator. So this big long thing here we're gonna put the coefficients and we're gonna start at the left, the coefficient corresponding to the highest exponent all the way down on the right to the coefficient corresponding to the constant term. So we're going to start with two corresponding to the seven zero corresponding to X to the six, three corresponding to X to the five zero corresponding to X to the four minus five corresponding to X cube zero corresponding to X squared zero corresponding to X. And then finally our constant term four. So now we can start with the division. So we're starting on the left and we just bring that to all the way down and we have a number at the bottom here, the numbers at the bottom are gonna multiply to this -1 on the left hand side. Okay? So we're gonna take 2 -1. That's gonna be -2 and we're gonna put that answer in the box to the right and just above. Now we have this column We have numbers in both spots. So we can add those together. We're gonna get 0 -2. That's going to give us -2. And now we have a new number at the bottom again. We're going to multiply the left -2 times -1. There's two. We can add the numbers in this column to give us five. We can do it again. Five times minus one is going to give us minus five, adding the numbers in the column, gives us minus five -5 times -1 is going to give us positive five Adding the numbers in the column. It's going to give zero zero times -1. Zero adding in the column, zero plus zero is zero again. Zero times minus one is going to be zero again and we have another column. It's gonna give us zero finally 0 times -1 is going to get a zero again. And this time when we add in the column we're gonna get four. Okay. Alright, so when we have synthetic division, this final number here, this four is going to be our remainder. We'll call it our and we have the rest of the numbers. Okay, so we can work from right to left or we can work from left to right whatever you want. So we had a polynomial of degree seven. We're dividing it by a polynomial of degree one. So that's going to reduce that polynomial by one degree. So we know we're gonna have a polynomial of degree six. So we can start from the left. Okay. With our coefficient of two and that's going to correspond to the highest exponent, which we know is going to be six. Okay. And then we can work our way to the right. So next we have minus two. Next to the five plus five X. To the four minus five X cubed. We have plus zero, X squared plus zero X plus zero. Okay, so you don't necessarily have to write these constants or the zero terms. I just wrote them just to be clear and then we have a remainder. Okay, so remainder is four and that is going to be divided by what we initially divided our polynomial by X plus one. So this is our solution, This is the quotient we get and that's going to correspond to solution B. Great. Thanks for watching guys. See you in the next video.

In Exercises 17–32, divide using synthetic division. (x^7+x^5−10x^3+12)/(x+2)

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem