Factoring Polynomials: How do I express the area and perimeter in factored form?

Question

Our topic is factoring polynomials, and I can't seem to solve this question:

Express the area and perimeter of the shaded region in factored form.

We've discussed how to solve for the perimeter given the area, although I really don't understand it. We're done with factoring using the common monomial factor, difference of two squares, perfect square trinomial, and general quadratic trinomial. We recently discussed factoring the sum and difference of two cubes.

How do I express the area and perimeter of the shaded region in factored form?

Thanks

Answer 1

How wide is the shaded region? If the length of the larger rectangle is $a^3$, and the length of the right is $b^3$...

The width of the shaded region is $a^3-b^3$

How tall is the shaded region?

I assume the picture is indicating that the height is $a^2+ab+b^2$

What is the area of a rectangle described as in terms of width and height?

width times height

And what is that in this case?

$(a^3-b^3)(a^2+ab+b^2)$

What is the perimeter of a rectangle described as in terms of width and height?

twice the sum of width and height

And what is that in this case?

$2(a^3-b^3 + a^2+ab+b^2)$

Now... can we express any of these above in a more compact way via factoring? That is debatable, but I do recognize at least one simplification you can do for area:

Knowing that $a^3-b^3=(a-b)(a^2+ab+b^2)$ we have that the area can be described as $(a-b)(a^2+ab+b^2)^2$. Is that really more preferred than $(a^3-b^3)(a^2+ab+b^2)$? I don't see a big difference between the two, so personally I wouldn't care which. They are the same to me.

Using the same for perimeter, we can do the following:

$2(a^3-b^3 + a^2+ab+b^2) = 2((a-b)(a^2+ab+b^2)+1(a^2+ab+b^2)) = 2(a-b+1)(a^2+ab+b^2)$. Again, is this more useful than the first way of writing it? That is debatable...They are not fundamentally different.