Find the quotient when 𝑥 to the six minus ten 𝑥 to the fourth plus sixteen 𝑥 cubed minus nine 𝑥 squared minus four 𝑥 plus fourteen is divided by 𝑥 squared plus two 𝑥 minus seven.
It’s a good idea if you’re missing a term to go ahead and add that in. So we were missing an 𝑥 to the fifth, which means it was zero 𝑥 to the fifth. So by adding that in, it will remind us that just in case we have an 𝑥 to the fifth term, we can put it above that zero 𝑥 to the fifth.
So our first step is to decide what we multiply to 𝑥 squared so that it’s equal to 𝑥 to the sixth. That would be 𝑥 to the fourth. And now we distribute. After distributing, now we subtract. So here we can see it was a good idea to add in that zero 𝑥 to the fifth, so we can combine it with that two 𝑥 to the fifth. We didn’t put anything above the zero 𝑥 to the fifth for our answer, but that’s okay.
And now we repeat the process. So what we multiply to 𝑥 squared so that it’s equal to negative two 𝑥 to the fifth. That would be negative two 𝑥 cubed. And now we distribute. Before we subtract, we need to bring down the sixteen 𝑥 cubed, and now we can subtract.
And now we repeat the process. So we will need to multiply 𝑥 squared by 𝑥 squared for it to be 𝑥 to the fourth. And now we distribute. And we also brought down the negative nine 𝑥 squared, and subtracting gives us negative two 𝑥 squared. So how do you get 𝑥 squared to look like negative two 𝑥 squared? You multiply by negative two. And negative two is a constant, so we put it above our constant. So if we would like to, we can add in a zero 𝑥.
And now we distribute. And we also need to bring down the negative four 𝑥 and the fourteen. And when we subtract, we get zero. So that means there is no remainder. So our final answer is 𝑥 to the fourth minus two 𝑥 cubed plus 𝑥 squared minus two.