Perform each division. See Examples 9 and 10. (k^4-4k^2+2k+5)/(k^2+1)

Question

Accepted Answer

Hello, today we're going to be performing long division on the given fraction. So what we are given is a to the fourth plus 13, a squared plus five, A plus 12 divided by A squared plus six. Now, before we set up the division, we always want to check to make sure whether we're missing any terms or not. And in our current numerator, we are missing one term, we're missing an A cubed term while we can go ahead and include the A cube term into the numerator by giving it a coefficient of zero. And when we set up the division, what we're going to be dividing is A to the fourth plus zero. A cubed Plus 13, a squared plus five a plus 12. And that is going to be divided by A squared plus six. Now, what we want to go ahead and do is multiply the quantity A squared plus six by a number that's going to allow the first term to match up as close as possible to the first term of the quotient. So here, what can we multiply A squared plus six by to get A squared as close to a to the power of four. Well, we can multiply that by A squared because a squared times A squared plus six is going to give us a to the fourth plus six A squared. And we're gonna go ahead and put that underneath this entire quantity. So again, lining up with respect to the power, we have a to the fourth plus six A squared. Now notice that we still don't have a cube term. So we're gonna go ahead and add the A cube term by giving it a coefficient of zero. Now you want to go ahead and be careful here because we do need to subtract this quantity. And since we're subtracting the quantity, we're going to need to distribute the negative sign throughout eight to the power of four plus zero A cubed plus six A and doing this is going to change the signs. So distribute the negative sign is going to give us negative eight to the fourth minus zero A cubed minus six A squared. And now we can go ahead and reduce A to the fourth minus eight to the fourth is just going to cancel each other out zero a cubed zero a cubed is not going to give us anything. So we don't have to worry about that term and 13 A squared minus six A squared is going to give us seven A squared. And we're gonna go ahead and bring down the next term which is five a and we're gonna repeat this process. Now, what's the number that I can multiply A squared plus six by to get A squared as close as possible to seven A squared. But we can multiply that quantity by positive seven because seven times A squared plus six is going to give us seven A squared plus 42. And so we're gonna go ahead and put that underneath this quantity, which is going to be 78 squared plus 42. Notice that we don't have an A term, so we can go ahead and add a term by giving it a coefficient of zero. And we don't have a constant value for the above quantity here. So we're gonna go ahead and bring down the 12 and that's where we're going to be simplifying. Now again, we want to go ahead and subtract this quantity. So we're going to need to distribute the negative sign throughout seven A squared plus zero A plus 42. And this is going to once again just change our signs. So what we have now is negative seven A squared minus zero A minus 42. And now we can reduce again, seven A squared minus seven A squared is just gonna cancel each other out. five A 0 A is going to give us five a And positive 12 -42 is going to give us -30. Now, there's no number that we can multiply A squared plus six by to get as close as possible to five A because the denominator has an exponent greater than the remainder. So five A minus 30 is going to be our remainder. So let's go ahead and put our quotient together. So we figured that we're going to have a squared plus seven. But how do we represent the remainder with this quantity? Well, five A is a positive term. So we're gonna go ahead and add the remainder to this quotient. And what goes in the denominator is going to be the original denominator, which is a squared plus six. And the numerator is going to have our current remainder, which is going to be five A minus 30. And this is going to be the quotient for the long division. And that means that the answer to this problem is going to be a. So I hope this video helps you understand how to do long division. And I'll go ahead and see you all in the next video.

Perform each division. See Examples 9 and 10. (k^4-4k^2+2k+5)/(k^2+1)

Verified Solution

Key Concepts

Polynomial Division

Synthetic Division

Remainder Theorem