Perform each division. See Examples 9 and 10. (4x^3+9x^2-10x-6)/(4x+1)

Question

Accepted Answer

Hello, today we're going to be simplifying the given quantity using long division. So what we are given is two K cubed plus three K squared minus 39 K minus 13. And we're going to be dividing this quantity by two K plus five. Now, before we set up the long division, you always want to check to make sure whether your numerator is missing any terms or not. But in this case, we don't have any missing terms. So we can go ahead and set the long division right away. So what we have is two K cubed plus three K squared minus 39 K minus 13. And we're going to be dividing this quantity by two K plus five. Now, in order to start this, you want to think to yourself, what is a number that I can multiply two K plus five by to get two K as close as possible to two K cubed? Well, I can multiply that quantity by K squared as two K plus five times K squared while you're gonna distribute K squared into two K plus five. And that's going to give us two K cubed plus five K squared. And we're gonna go ahead and line this quantity up in the bottom. So what we have is two K cubed plus five k squared. And you want to be careful here because we want to go ahead and subtract this quantity. So since we're subtracting this quantity, we're going to need to distribute the negative sign into two K cubed plus five K squared. And doing this is going to change the signs. So what we now have is negative two K cubed minus five K squared. And we can go ahead and continue the simplification, two K cubed minus two K cubed is going to cancel itself out and three K squared minus five K squared is going to give us negative two K squared. We bring down the next term which is going to be negative 39 K and we want to go ahead and repeat this process. Now again, what's the number that I can multiply two K plus five by to get two K as close as possible to negative two K squared. While we can multiply this quantity by negative K, two K plus five multiplied by negative K. While you distribute negative came to two K plus five, that's going to give us negative two K squared minus five K. So again, you put this quantity on the bottom negative two K squared minus five K and we're going to be subtracting this quantity. So we're going to need to distribute the negative sign into negative two K squared minus five K doing this is going to change the values of the sign. And what we now have is positive two K squared plus five K, negative two K squared plus two K squared is going to cancel itself out and negative 39 K plus five K is going to give us negative 34 K. We bring down the next term which is going to be negative 13 and repeat the process one more time. Now, what's the number that I can multiply two K plus five by to get me as close as possible to negative 34 K. While we can multiply this quantity by negative two K plus five multiplied by negative is going to give us -34 K minus 85. So we have negative 34K -85. And once again, we want to go ahead and distribute the negative sign to this quantity since we want to subtract it. So doing this is going to change the values of the signs as now we have K plus 85 negative 34 K plus K cancel itself out and negative 13 plus 85 is going to give us positive 72 this is going to be our remainder. So let's go ahead and put her answer together. What we calculated for the quotient was K squared minus K minus 17 and we want to go ahead and include the remainder into this quotient. The remainder is positive 72. So we're going to be adding the remainder and we're going to represent the remainder as a fraction where the value of the remainder goes in the numerator. So we're going to be putting 72 in the numerator. And the original denominator goes in this denominator, which is going to be two K plus five. And this is going to be the simplified form of the given quantity using long division. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to use 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+9x^2-10x-6)/(4x+1)

Verified Solution

Key Concepts

Polynomial Division

Synthetic Division

Remainder Theorem