Perform each division. See Examples 9 and 10. (3x^3-2x+5)/(x-3)

Question

Accepted Answer

Hello, today we're going to be simplifying the given quantity by using long division. So what we are given is five a cubed minus A plus nine divided by a minus two. Now, before you set up the long division, you always want to check to make sure whether you're missing any terms in the numerator or not. And here we're missing one term. We don't have an A square term while we can include the a square term into our numerator by giving it a coefficient of zero. So when we set up the long division, we're going to have five A cubed plus zero A squared minus A plus nine and we're going to be dividing this quantity by a minus two. Now, the way that you continue this is you want to ask yourself, what is a number that I can multiply A minus two by to get a as close as possible to five A cubed. Well, what if I multiply that quantity by five A squared Multiplying a -2 by five A squared while we're going to need to distribute five A square into a minus two and that's going to give us five a cubed minus 10 A squared. So we're gonna go ahead and put this quantity underneath the quotient. And so this is going to give us five a cubed minus 10 A squared. Now, you want to be careful with your setup because you want to subtract this quantity. And since we're subtracting the quantity, we're going to need to distribute the negative sign into five A cubed minus 10 A squared. And doing this is going to change the signs of this quantity. So what we now have is negative five A cubed plus 10 A squared. And we can go ahead and continue with the simplification, five A cubed minus five A cube is just going to cancel each other out and zero A squared plus 10 A squared is going to give us A squared. We bring down the next term which is negative A and we want to repeat this process. Now, what's a number that I can multiply A minus two by to get a as close as possible to 10 A squared. Well, I can multiply this by positive 10 A because a minus two times 10 A while we're going to distribute 10 A two A minus two. And this is going to give us 10 A squared minus A. So we're gonna put this quantity underneath -20 a. And again, we want to subtract this quantity. So we're going to need to distribute a negative sign and distributing the negative sign is gonna once again change our signs. And so what we now have is negative 10 A squared plus 28 10 E squared minus 10 squared is going to cancel each other out And negative a plus 28 is going to give us positive 19 a. Once again, we bring down the next term which is going to be positive nine and we're going to repeat this process one more time Now again, what is a number that I can multiply a minus to buy to get a as close as possible to 19 a while I can multiply this quantity by positive 19 As a -2 times 19 is going to give me 19 a minus 38. So we're gonna put this underneath 19 A minus we want to subtract this quantity. So again, we're going to need to distribute the negative sign and doing this is going to change our signs. So what we now have is negative 19 8 plus 38 19 A minus 19 A is going to cancel each other out and 38 plus nine is going to give me positive 47 this is going to be our remainder. So let's go ahead and put our answer together by first writing the whole question that we came up with what we calculated was five A squared plus A plus 19. And we're going to need to include the remainder in this quantity while the remainder is 47 which is a positive number. So we're going to be adding the remainder to this quantity. And we're going to represent the remainder as a fraction where the value of the remainder goes in the numerator. So 47 is in the numerator and the original denominator stays in this denominator which is going to be a -2. And this is going to be the simplified form of the quantity given to us 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 perform long division. And I'll go ahead and see you all in the next video.

Perform each division. See Examples 9 and 10. (3x^3-2x+5)/(x-3)

Key Concepts

Polynomial Division

Synthetic Division

Remainder Theorem