Question

Factor the expression given below. Write each factor as a polynomial in

descending order. Enter exponents using the caret (^). For example, you
would enter x2 as x^2.
343x3 + 216y3
Answer here

Answer 1

Explanation:

This is the sum of perfect cubes. There is a pattern that can be followed in order to get it factored properly. First let's figure out why this is in fact a sum of perfect cubes and how we can recognize it as such.

343 is a perfect cube. I can figure that out by going to my calculator and starting to raise each number, in order, to the third power. 1-cubed is 1, 2-cubed is 8, 3-cubed is 27, 4-cubed is 64, 5-cubed is 125, 6-cubed is 216, 7-cubed is 343. In doing that, not only did I determine that 343 is a perfect cube, but I also found that 216 is a perfect cube as well. Obviously, x-cubed and y-cubed are also both perfect cubes. The pattern is

(ax + by)(a^2x^2 - abxy + b^2y^2) where a is the cubed root of 343 and b is the cubed root of 216. a = 7, b = 6. Now we fill in the formula:

(7x + 6y)(7^2x^2 - (7)(6)xy +6^2y^2) which simplifies to

(7x + 6y)(49x^2 - 42xy + 36y^2)