In Exercises 1–16, divide using long division. State the quotient...

Question

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^4−81)/(x−3)

Accepted Answer

Hey everyone here we are asked to find the quotation of the expression using long division. So setting this up in long division formatting to the left, we will have our divisor which is x minus six and to the right underneath the bar will have the dividend. And for clarity I'm going to include placeholders for the missing powers. So we have X. To the fourth power plus zero X cubed plus zero X squared plus zero X minus 1296. Now to start solving we begin by taking the leading term of the dividend and dividing it by the leading term of the divisor. So we have X to the fourth power divided by X which simplifies to X cubed. And now we take this term and we add it to the top of the bar and it becomes our first term of our quotient. We also take this term and multiply it by the divisor. So we have x cubed times the quantity of x minus six. Multiplying we get X to the fourth power minus six X cubed. We then take this resultant and subtract it from our dividend. So we have minus the quantity of X to the fourth power minus six X cubed. Now subtracting we get X to the fourth power minus X to the fourth power which is just zero then zero X cubed minus negative six X cubed gives us positive six X cubed. Now just bringing down the rest of the terms, we also have plus zero X squared plus zero X minus 1296. So now we repeat this process but we now take the leading term of the remainder and again divided by the leading term of the divisor. So we have six X cubed divided by X. Which simplifies to six X squared. Now we move this term to the top of the bar so our second term of the quotient becomes six X squared. We then take this quiet by our divisor. So we have six X squared times the quantity of x minus six. Now multiplying this, we get six X cubed minus 36 X squared. And now we subtract this resultant from the remainder. So we have minus the quantity of six X cubed minus 36 X squared. Subtracting we get six X cubed minus six X cubed. Which gives zero then zero X squared minus negative 36 X squared gives positive 36 X squared. And now we can just bring down the last two terms. So we have plus zero x minus 1296. And now we repeat the process again and we take this 36 X squared and divided by X. This applies to 36 X. We then take the 36 X. And added to the top of the bar, becoming our third term of the quotient. And from here we multiply 36 36 X by our divisor. So we get 36 X times the quantity of x minus six which is 216 x. From here. We subtract this resultant from the remainder. So we have minus the quantity of 36 X squared minus 216 X. Subtracting we see that the x squares cancel. So we have zero and then we have zero x minus negative 216 X. Which gives us positive 216 X. We then just bring down this last term. So we also have minus 1296. We then take repeat the process and take the leading term here which is 216 x. And we divided by the leading term of the divisor which is X. And this gives us just 216 Term of the Quotient becomes 216. We then take this value and multiply it by our divisor. So we have 216 times the quantity of X -6. Multiplying. We get X -1296. We then subtract this resulted from our remainder. So we have minus the quantity of 216 X -1296. 216 x minus 216 X is just zero and negative. 1296 minus negative. 1296 is also just zero. So now we can see that the degree of the remainder is less than the degree of the divisor. So we are done with this process. We now see also that our remainder is zero from this. We cannot clearly see that our quotient our final answer is X cubed plus six X squared plus 36 X plus 216. And this is our answer here for choice. C. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^4−81)/(x−3)

Key Concepts

Polynomial Long Division

Quotient and Remainder Theorem

Degree of a Polynomial