Knowing that the length of a rectangle is three 𝑥 minus four and its area is six 𝑥 to the fourth minus eight 𝑥 cubed plus nine 𝑥 squared minus nine 𝑥 minus four, express the width of the rectangle as a polynomial in standard form.
Here we have a rectangle, its length is equal to three 𝑥 minus four. We are solving for the width. And then our area is equal to this polynomial. Length times width is equal to the area. So if we are solving for the width, which will be 𝑊, we need to divide both sides by 𝐿. This means, in order to find the width, we need to take the area and divide by the length.
So here we can solve by dividing, and we can use something called long division. So we begin by looking at the first terms. What do we multiply to three 𝑥 so it looks like six 𝑥 to the fourth? We will need to multiply it by two 𝑥 cubed, so we put that above the cubed term. And now we distribute. And now we subtract. So all of these cancel. So this means we need to bring down the nine 𝑥 squared. And now we start over.
How do we get three 𝑥 to look like nine 𝑥 squared? What do we multiply by? And that would be three 𝑥, we put above the 𝑥 term. We can always add in a zero for something that isn’t being used. So for here, that’ll be zero 𝑥 squared. We do not need to include it in our final answer. So now we distribute the three 𝑥. Now after we’ve distributed, we need another term, so we bring down the negative nine 𝑥. And now we subtract. So we have negative nine 𝑥 minus negative 12 𝑥, so it’s really plus 12 𝑥. So we get three 𝑥, and we start the process over again.
How do we get three 𝑥 to look like three 𝑥? What do we multiply by? And that would be one. So we distribute, bring down the minus four, and subtract. And we get zero. So this means our remainder is zero.
So this means our width is two 𝑥 cubed plus three 𝑥 plus one.