Question Video: Finding a Quotient and Remainder Using Polynomial Division Mathematics

Video Transcript

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

To solve this problem, I’m actually gonna set up a long division. Okay, so when we look at this, I’ve set up the long division that we’re going to complete. So we can see that we’re gonna divide the two 𝑥 to the power of four plus three 𝑥 cubed minus five 𝑥 minus five all by two 𝑥 minus one. It’s worth noting here that I’ve actually put in a value here of zero 𝑥 squared.

Now, the reason I’ve done this is actually to keep everything aligned. So it doesn’t matter if you haven’t got one particular power of 𝑥. I’d always recommend putting in even zero 𝑥 squared or zero 𝑥 cubed — whichever power of 𝑥 you don’t have — just because it keeps everything aligned to make sure that we don’t make any mistakes when we’re 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 𝑥 minus one is two 𝑥. So what we want to see is how do we get to two 𝑥 to the power of four using two 𝑥. What would I have to multiply it by? But we can see that actually two 𝑥 we’d have to multiply it by 𝑥 cubed to give us two 𝑥 to the power of four. So therefore, the first term in our quotient is actually gonna be 𝑥 cubed.

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

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

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

So now, we’re 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 𝑥 from our term two 𝑥 minus one by to get to four 𝑥 cubed. Great! So we can now work out that two 𝑥 actually needs to be multiplied by two 𝑥 squared to get to four 𝑥 cubed because we got two multiplied by two, which gives us four, and that 𝑥 multiplied by 𝑥 squared gives us the 𝑥 cubed.

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

So first of all, we’re going to do the two 𝑥, which gives us four 𝑥 cubed. Now, we’re gonna multiply our two 𝑥 squared by negative one, which gives us negative two 𝑥 squared. Then yet again, we’re gonna subtract our powers of 𝑥 like we did previously. So again, we get four 𝑥 cubed minus four 𝑥 cubed would give us zero. So we don’t need to write anything there.

But then, for the next part, we’re gonna have zero 𝑥 squared minus negative two 𝑥 squared. So remembering as we did before, that means we’re gonna add two 𝑥 squared. So actually, it’s going to give us an answer of two 𝑥 squared. And again, we brought down our next term, which is our 𝑥 term. So we’ve got two 𝑥 squared minus five 𝑥.

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

So now we’re going to subtract again. So as previously, the two 𝑥 squared minus two 𝑥 squared is just zero. So we don’t have to worry about that. But then, we’ve got negative five 𝑥 minus negative 𝑥, which is gonna give us negative four 𝑥. Remember being careful of the minus and negative. So it’s negative five 𝑥. And then, we’re actually going to add on an 𝑥, which gives us negative four 𝑥. And then, we bring our final term down. So we get negative four 𝑥 minus five.

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

So then, we do our final round of subtraction. And again, negative four 𝑥 minus negative four 𝑥 would just be equal to zero because it’s like negative four 𝑥 add four 𝑥. But then, we’re 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 𝑥 to the power of four plus three 𝑥 cubed minus five 𝑥 minus five by two 𝑥 minus one. So therefore, we can say that our quotient, so 𝑞𝑥, is equal to 𝑥 cubed plus two 𝑥 squared plus 𝑥 minus two. And our remainder 𝑟𝑥 is equal to negative seven.