### Video Transcript

Find the remainder ๐ ๐ฅ and the quotient ๐ ๐ฅ when two ๐ฅ to the power of four plus three ๐ฅ cubed minus five ๐ฅ minus five is divided by two ๐ฅ minus one.

To solve this problem, Iโm actually gonna set up a long division. Okay, so when we look at this, Iโve set up the long division that weโre going to complete. So we can see that weโre gonna divide the two ๐ฅ to the power of four plus three ๐ฅ cubed minus five ๐ฅ minus five all by two ๐ฅ minus one. Itโs worth noting here that Iโve actually put in a value here of zero ๐ฅ squared.

Now, the reason Iโve done this is actually to keep everything aligned. So it doesnโt matter if you havenโt got one particular power of ๐ฅ. Iโd always recommend putting in even zero ๐ฅ squared or zero ๐ฅ cubed โ whichever power of ๐ฅ you donโt have โ just because it keeps everything aligned to make sure that we donโt make any mistakes when weโre going to the next stage.

Well, first part of our division that we need to look at is we need to look at these terms here. So our first term, on the two ๐ฅ minus one is two ๐ฅ. So what we want to see is how do we get to two ๐ฅ to the power of four using two ๐ฅ. What would I have to multiply it by? But we can see that actually two ๐ฅ weโd have to multiply it by ๐ฅ cubed to give us two ๐ฅ to the power of four. So therefore, the first term in our quotient is actually gonna be ๐ฅ cubed.

The next step in the question is actually to multiply this first term of our quotient ๐ฅ cubed by both terms in two ๐ฅ minus one. So if we multiply two ๐ฅ by ๐ฅ cubed, we get two ๐ฅ to the power of four. And then if we multiply negative one by ๐ฅ cubed, weโre gonna get negative ๐ฅ cubed. So weโve now written this and making sure that actually all our powers of ๐ฅ are aligned.

So actually, the next stage is now to subtract our powers of ๐ฅ. So first of all, we have two ๐ฅ to the power of four minus two ๐ฅ to the power of four. Thatโs gonna give us zero. And then next, we have three ๐ฅ cubed and then minus negative ๐ฅ cubed. Just be careful here because actually a lot of the common mistakes here would be to subtract the ๐ฅ cubed from three ๐ฅ cubed to give us two ๐ฅ cubed. But in fact as I said, itโs three ๐ฅ cubed minus negative ๐ฅ cubed. So weโre gonna add them. So it gives us four ๐ฅ cubed.

Okay, for the next stage, we are now gonna bring down the next power of ๐ฅ to our next term. And this is where it was important that we put in the zero ๐ฅ squared so that actually itโs keeping everything aligned. So weโve now actually got four ๐ฅ cubed plus zero ๐ฅ squared. I can ignore the zero plus at the beginning. So Iโve just removed that to tidy up.

So now, weโre gonna do the same thing which as we did at the beginning. We can try and see now what you have to multiply the two ๐ฅ from our term two ๐ฅ minus one by to get to four ๐ฅ cubed. Great! So we can now work out that two ๐ฅ actually needs to be multiplied by two ๐ฅ squared to get to four ๐ฅ cubed because we got two multiplied by two, which gives us four, and that ๐ฅ multiplied by ๐ฅ squared gives us the ๐ฅ cubed.

Okay, great! So we now know the next part of our quotient, which is going to be two ๐ฅ squared. So we now got ๐ฅ cubed plus two ๐ฅ squared. And again, now at this stage, weโre actually gonna multiply the two ๐ฅ and the negative one both by our two ๐ฅ squared.

So first of all, weโre going to do the two ๐ฅ, which gives us four ๐ฅ cubed. Now, weโre gonna multiply our two ๐ฅ squared by negative one, which gives us negative two ๐ฅ squared. Then yet again, weโre gonna subtract our powers of ๐ฅ like we did previously. So again, we get four ๐ฅ cubed minus four ๐ฅ cubed would give us zero. So we donโt need to write anything there.

But then, for the next part, weโre gonna have zero ๐ฅ squared minus negative two ๐ฅ squared. So remembering as we did before, that means weโre gonna add two ๐ฅ squared. So actually, itโs going to give us an answer of two ๐ฅ squared. And again, we brought down our next term, which is our ๐ฅ term. So weโve got two ๐ฅ squared minus five ๐ฅ.

Yet again, we complete the step when we look at- well, two ๐ฅ multiplied by what gives us two ๐ฅ squared. Well, itโs going to be two ๐ฅ multiplied by ๐ฅ gives us two ๐ฅ squared. So again, we can add this up as the next term in our quotient. And again, weโre gonna multiply two ๐ฅ minus one each term by our new term in the quotient, which is ๐ฅ. So we get two ๐ฅ squared when we multiply two ๐ฅ by ๐ฅ. Then weโre gonna get negative ๐ฅ when we multiply negative one by ๐ฅ. So weโre left with two ๐ฅ squared minus ๐ฅ.

So now weโre going to subtract again. So as previously, the two ๐ฅ squared minus two ๐ฅ squared is just zero. So we donโt have to worry about that. But then, weโve got negative five ๐ฅ minus negative ๐ฅ, which is gonna give us negative four ๐ฅ. Remember being careful of the minus and negative. So itโs negative five ๐ฅ. And then, weโre actually going to add on an ๐ฅ, which gives us negative four ๐ฅ. And then, we bring our final term down. So we get negative four ๐ฅ minus five.

And for the final time, we have to say again two ๐ฅ multiplied by what gives us our negative four ๐ฅ. So we can see that actually itโs gonna be two ๐ฅ multiplied by negative two to give us negative four ๐ฅ. So that means our final term in our quotient is negative two. And then, we multiply this negative two by both terms in two ๐ฅ minus one. So weโve got two ๐ฅ multiplied by negative two, which gives us negative four ๐ฅ. And then, we have negative two multiplied by negative one, which gives us positive two. So weโre left with negative four ๐ฅ plus two.

So then, we do our final round of subtraction. And again, negative four ๐ฅ minus negative four ๐ฅ would just be equal to zero because itโs like negative four ๐ฅ add four ๐ฅ. But then, weโre gonna do negative five minus two, which gives us our final answer of negative seven. So then that means we have now fully divided two ๐ฅ to the power of four plus three ๐ฅ cubed minus five ๐ฅ minus five by two ๐ฅ minus one. So therefore, we can say that our quotient, so ๐๐ฅ, is equal to ๐ฅ cubed plus two ๐ฅ squared plus ๐ฅ minus two. And our remainder ๐๐ฅ is equal to negative seven.