Hi John,

This follows from what is called "the distributive property of multiplication over addition". The property is just this

2 (3 + 5) = 2 8 =16

but there is another way to evaluate this expression, you can distribute the multiplication over the addition and then add. What I mean is

2 (3 + 5) = 2 3 + 2 5 = 6 + 10 = 16

The distributive property is that

a (b + c) = a b + a c

This distribution also is valid if the multiplication is on the right

(a + b) c = a c + b c

Now let's look at your problem

(a - b)² = (a - b) (a - b)

I want to see the first "(a - b)" as one quantity and call it x, and I also want to see the second "(a - b)" as (a + (-b)). The expression is then

(a - b)² = (a - b) (a - b) = x (a + (-1)b)

The distributive property then says that this is

x a + x (-1)b = x a - x b = (a - b) a - (a - b) b

Now use the distributive property on the right (again seeing (a - b) as (a + (-1)b).

(a - b) a - (a - b) b = a a - a b - (a b - b b)
= a² - ab - (ab - b²)
= a² - 2 ab - b²

Penny

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