Why Study Algebra? - Special Products - The Sum and Difference of Two Cubes
Posted: Monday, May 02, 2011
by Joe Pagano
Math by Joe
There are two special products in algebra that are worthy of mentioning: the sum and difference of two cubes. Although the quadratics are much more common, the cubes and indeed higher order polynomials find their place in all sorts of interesting applications. For this reason, learning how to factor x^3 + y^3 and x^3 - y^3deserve some attention. Let us explore them here.
For the difference case, that is x^3 - y^3, we factor this as (x - y)(x^2 + xy + y^2). Notice the signs correspond in the first factor but all signs in the second are postive. This too, is the key to remembering the factorization. Just remember that for the difference case, the first factor is (x - y) and that the second factor hasall positives. This insures that the cross terms cancel and we are left with just x^3 - y^3.
To understand the exposition above, let us actually multiply the sum case out (the difference case is entirely similar). We have
(x + y)(x^2 - xy + y^2) = x(x^2 -xy + y^2) + y(x^2 -xy + y^2). Notice how I have used the distributive property tosplit this multiplication. After all, that is what this property does. Now the first yields x^3 - x^2y + xy^2 and the second yields, x^2y - xy^2 + y^3. (Observe the use of the commutative property of multiplication, that is yx^2 = x^2y). Adding the two pieces together, we have -x^2y + xy^2 cancel with x^2y - xy^2. Thus all we are left with isx^3 + y^3.
What becomes a bit more challenging is the factoring of a perfect cube and a number, which is also a perfect cube. Thus x^3 + 8. If we write this as x^3 + 2^3, we see that we can factor this into (x + 2)(x^2 - 2x + 4). If we think of 2 as y, then we see this exactly corresponds to what we have just done. To make sure this is crystal clear, consider x^3 - 27. Since 27 is equal to 3^3, and is therefore a perfect cube, we can apply what we just learned and write x^3 - 27 = x^3 - 3^3 = (x - 3)(x^2 + 3x + 9).
From now on, when you see cubes, think perfect cubes, and don't forget that you can always factor their sum or difference into a product using the rules we just discussed. Maybe if you get past this hurdle in algebra, you just might be heading for the finish line. Till next time...
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