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

Question

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

Accepted Answer

Hey everyone welcome back in this video. We're gonna be using synthetic division to find the quotient of the following polynomial were given X to the five plus three X cubed minus six. All divided by x minus three. The first thing we want to do with synthetic division is to take a look at the denominator or the divisor and be sure that it's a degree one polynomial in this case. Our divisor is x minus three. So the highest exponent is one. That means degree one polynomial. So we're okay to go ahead. Now with synthetic division, we're going to take X to the five plus three X cubed. The numerator that were given. And we're just going to expand it. We're gonna want to use the coefficient of every single term from the highest exponents. All the way down to the constant term. We're just going to expand and include the terms that are missing. So X to the five plus three X cubed minus six becomes X. To the five plus zero X. To the four plus three X cubed plus zero, X squared Plus zero, X -6. Okay, so again, that's our numerator. That's the same polynomial. We've just written it out including those zero terms. All right now let's draw our synthetic division table and get started. The first number. We're going to add to our table is going to be the number on the left. And in this case that's going to be three. Alright, now, where does that come from? Well it comes from the denominator or the divisor. Which in this case is X -3. What we do is we essentially set x minus three equal to zero. We solve for X. We get X equals 23. And that's the number that we're using here on the left of our table. Okay. So it's essentially like the root of the denominator. Alright, so now to complete the rest of the top row in our synthetic division table, we're gonna use the coefficients of the expanded polynomial from the numerator X to the five plus zero X to the four plus three X cubed plus zero, X squared plus zero, X minus six. So we're gonna use the coefficients in descending order from highest exponent down. So we have one corresponding to X to the 50 corresponding to X. The 43 corresponding to X cubed zero corresponding to x squared zero corresponding to X and minus six. The constant term. Okay, so to get started we're gonna take this one on the far left. We're going to bring it down and rewrite it at the bottom. Okay? It stays the same. Now when we have numbers at the bottom they're gonna multiply to this three on the left. So we're gonna take one times three gives us three. We're just gonna write it to the right and above. We're going to add numbers in the call in columns in the table. So we're gonna take zero plus three. That's going to give us three. We're gonna write it down below. Okay now we have a new number at the bottom and then we're going to multiply it on the left. So three times three that gives us nine. We read it to the right and above. We add numbers within the column. Three plus nine is going to give us 12, Write it down below and now we take the number at the bottom. 12. We multiply it by the three on the left. That gives us 36. We add in the column. zero plus 36 is 36. Again we multiply 36 times three. It's gonna give us 108 adding in the column, Gives us 108 And for one final time we take the 108 at the bottom. We multiply it by the three on the left. That gives us 324 and then we add in the column. So we have minus six plus 324. It's going to give us 318. Okay. Alright. So we've done the synthetic division. Now we just need to interpret the result. So the first thing is this number, the last number on the far right? 318 here is going to be our remainder. We can call it are. Okay. Alright now we're writing out the solution, we want to think about what the highest exponent will be. Well the problem gave us a polynomial of degree five X. To the five was the highest term in the numerator. And we're dividing by a polynomial of degree one, X minus three. So when we're dividing by a polynomial of degree one we're going to reduce the highest exponent by one Power. Okay so our new polynomial or a new function, the highest order will be four. So we're gonna start with the X. to the four. And the coefficients for this function are gonna come from our synthetic division table. Okay so the first coefficient is one, so X to the four is gonna have coefficient one. Okay. And we're gonna work down from X to the four down to the constant term. And as we work to the right we're gonna take the coefficients from our table. So next we're gonna have three. The coefficient of three from our table, that's going to correspond with the X cubed term. We're gonna have 12. The next coefficient from our table corresponds to X squared 36. The next coefficient from our table corresponds to the linear term and finally 108 is going to be our constant term. Alright. And lastly we just need to write the remainder. The remainder is going to be 318 and that has to be divided by the original divisor X -3. Okay, because it's left over. Alright, so there's our solution to the division and that's going to correspond with answer A Alright thanks everyone. See you in the next video.

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

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem