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Video: Simplifying Polynomials by Division to Express the Width of a Rectangle

Bethani Gasparine

Knowing that the length of a rectangle is 3𝑥 − 4 and its area is 6𝑥⁴ − 8𝑥³ + 9𝑥² − 9𝑥 − 4, express the width of the rectangle as a polynomial in standard form.

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Video Transcript

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.

The area of a rectangle is length times width. And if we know the area and we know the length, we can divide them and solve for the width. So we can use polynomial division.

So first we decide what do we have to multiply three 𝑥 by in order for it to look like six 𝑥 to the fourth. That would be two 𝑥 cubed, so we put it above the 𝑥 cubed term. And now we distribute, and we’re subtracting, and they actually cancel. So we bring down our nine 𝑥 squared. So how do we get three 𝑥 to look like nine 𝑥 squared? We multiply by three 𝑥, so we put it above the 𝑥 term. And we could add in a plus zero 𝑥 squared above the 𝑥 squared term, if we wanted to.

So now we distribute. We need to bring down the negative nine 𝑥. This way, we can subtract the negative 12x from something and we get three 𝑥. So what do we multiply three 𝑥 by to look like three 𝑥? That would be one, bringing down the negative four and then subtracting, we get zero. So that means we don’t have a remainder. So our final answer is that the width is two 𝑥 cubed plus three 𝑥 plus one.