Explain how the distributive property helps us multiply the following polynomials and why and how the final products differ: (a + b)^2, (a – b)^2, and (a - b)(a + b).

Question

Explain how the distributive property helps us multiply the following polynomials and why and how the final products differ: 

1. (a + b)^2, 
2. (a – b)^2, and 
3. (a - b)(a + b).

Answer 1

The FOIL method is a version of the distributive property, twice applied

(a+b)^2 =(a+b)(a+b) = (a+b)(a) + (a+b)(b) = a^2 +ab + ab + b^2 = a^2 +2ab +b^2

same with

(a-b)^2 = (a-b)(a-b) = (a-b)a + (a-b)(-b) = a^2 -ab -ab +b^2 = a^2 -2ab +b^2

or

(a-b)(a+b) = (a-b)a + (a-b)(b) = a^2 -ba +ab -b^2 = a^2 -b^2