Perform each division. See Examples 9 and 10. (4x^3-3x^2+1)/(x-2)

Question

Accepted Answer

Hello, today we're going to be simplifying the given quantity using long division. So what we are given is five T cubed minus 11 T squared plus four divided by T minus one. Now, before you set up your long division, you always want to check to make sure whether you're missing any terms from the numerator. And here we're missing exactly one term for T, we don't have a T to the power of one term, but we can include that term by giving it a coefficient of zero. So when you set up the long division, what you're going to have is five T cubed minus 11 T squared plus zero T Plus four. And you're going to be dividing this quantity by T -1. Now, in order to start this process, you want to go ahead and ask yourself what is the value that you can multiply T minus one to get T as close as possible to five T cubed. Well, what we can multiply this by is positive five T squared because if we multiply T minus one by five T squared, we distribute five T squared into T minus one. And that's going to give us five T cubed minus five T squared. So we're gonna put this quantity underneath and this is going to give us five T cubed minus five T squared. Now, you want to go ahead and be careful because we need to subtract this quantity. And since we're going to subtract this quantity, we're going to need to distribute the negative sign into five T cubed minus five T squared. And doing this is going to change the signs of this quantity. So what we now have is negative five T cubed plus five T squared. And now we can continue with the simplification. So five T cubed minus five T cubed is going to council itself fell and negative 11 T squared plus five T squared is going to give us negative six T squared. We're going to bring down the next term which is zero T and we're going to continue this process. Now, what's the value that I can multiply T minus one by to get T as close as possible to negative 60 squared. Well, we can multiply this quantity by negative 60 because T minus one multiplied by negative 60. We're going to distribute this into T minus one and that's going to give us negative 60 squared plus 60. We're going to put this quantity underneath negative 60 squared plus 60. And again, we're subtracting this quantity. So we're going to need to distribute the negative sign into negative 60 squared plus which is going to change the values of the signs. So distributing the negative sign is going to give us positive 60 squared minus 60 and negative 60 squared plus squared is going to cancel itself out and zero T minus 60 is going to give us negative 60. We bring down the next term which is positive four and we're going to repeat this process one more time. So what is the value that it can multiply T minus one by to get T as close as possible to negative 60 while we can multiply this by negative six as T minus one multiplied by negative six is going to give me negative 60 plus six. So again, we're gonna put this underneath And we're going to distribute a negative sign into this quantity doing this is going to change the values of the signs. So what we now have is positive -6 and negative 16 plus positive 16 is going to cancel itself out And 4 -6 is going to give us -2. So now we can go ahead and write the quotient together. So what we calculated was five T squared minus 60 minus six. And we need to include the remainder into this quotient negative two is our current remainder. So the remainder is going to be a negative sign or negative value with the actual value in the numerator which is going to be too and we're going to keep the same denominator as the original quantity, which is going to be T minus one. And this is going to be the simplified form of the original quantity using long division. And with that being said, the answer to this problem is going to be be so I hope this video helps you in understanding 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. (4x^3-3x^2+1)/(x-2)

Key Concepts

Polynomial Division

Synthetic Division

Remainder Theorem