Algebra: Common Products
In this algebra video tutorial, I demonstrate three polynomial product formulas and how to derive them. First, I derive the General Product formula for two binomials, and then I use it to derive the last two formulas. After each derivation, I apply each formula by making a polynomial substitution.
All of our product formulas for polynomials necessarily rely on the distributive property. As we will see in the derivation of the general formula, this is all that is needed for it. Below, the derivation is shown, which involves one application of the right-distributive property followed by two applications of the left-distributive property.
The second and third formulas, above, are simply specializations of the general formula. For clarity, I have called them the Square of a Sum and the Difference of Squares formulas so that I can refer to them by name. As I showed in the video, these can be derived from the general formula directly.
Beyond the derivations, it is important to understand how these formulas are applied in algebra. Below, the general formula is applied via substitutions for each of the variables: a = -3xy, b = 2y, c = -1, and d = x. The steps of the simplication are straight-forward and it is a good idea to try to supply the properties that are used in the simplification.
As a final point, I want to mention that the terms in the final step of the above application are written so that the monomials with the largest degree are first. This is a convention that is used in mathematics and it is good to follow. Also, the terms are typically written in alphabetical order, so that the x terms come before the y terms and so on. These conventions help us to easily compare polynomials quickly.