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+2x^3−4x^2−5x−6)/(x^2+x−2)

Accepted Answer

hey everyone here we are asked to find the quotient of the expression. Using long division. So setting this up using long division formatting, we need to set the dividend underneath the bar. So we have X to the fourth power plus 13 X cubed plus 37 X squared minus 26 X minus 40. And on the left hand side we need to include our divisor which is X squared plus seven X minus nine. And now to start solving we begin by taking the leading term of our dividend and dividing it by the leading term of our divisor. So we have X to the fourth power divided by X squared. Now we're calling the product rule exponents as well as the rule for negative exponents. This expression simplifies to X squared. Now we take this X squared and place it at the top of the bar. So it becomes our first term of our quote ship. We then take the X squared and multiply it by our divisor. So we have X squared times E quantity of X squared plus seven x minus nine. Now multiplying, we get X to the fourth power plus seven X cubed minus nine X squared. And next we take this resultant here and subtracted from our dividend. So we have minus the quantity of X to the fourth power plus seven X cubed minus nine X squared. Now subtracting we have X to the fourth power minus X to the fourth power which gives us zero then 13 X cubed minus seven X cubed gives us positive six X cubed. And next 37 X squared minus negative nine X squared gives us positive 46 X squared. Now we just bring down the last two terms. So we also have minus 26 X and minus 40. Now we just repeat the 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 squared which simplifies to six X. Again we bring this term to the top of the bar so six X becomes our second term of the quotient and next we also take the six X and multiply it by the divisor. So we have six x times the quantity of X squared plus seven X minus nine, multiplying this out. We get six X cubed plus 42 X squared minus 54 x. Now we just take this resulted and subtract it from our remainder. So we have minus the quantity of six x cubed plus 42 X squared minus X. Next subtracting we have six X cubed minus six X cubed which just gives us zero then 46 X squared minus 42 X squared gives us positive four x squared. Next negative 26 X minus negative X gives us positive 28 X. And we just have to bring down our last term. So we also have minus 40 repeating this process again we take the leading term which is four X squared and divided by the leading term of the divisor which is X squared. So we have four X squared divided by X squared. And the X's cancel out. So we are left with just four. Next we add four to the top and it becomes our third term of the quotient. We also take four and multiply it by the divisor. So we have four times the quantity of X squared plus seven X minus nine. Multiplying we get four X squared plus 28 X minus 36. Next we subtract this resultant from the remainder. So we have minus the quantity of four X squared plus 28 X minus 36. Now subtracting we have four X squared minus four X squared which is just zero then 28 X minus 28 X is also zero and finally negative 40 minus negative 36 gives us negative four. And now we can see that the degree of the remainder is less than the highest degree of the divisor. So we are done dividing And now this tells us that we also have a remainder of -4. Therefore our quotient. Our final answer is X squared plus six X plus four minus four divided by the quantity of X squared plus seven X minus nine. And this leaves us with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time

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In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^4+2x^3−4x^2−5x−6)/(x^2+x−2)