Algebra Know-It-ALL

Multiplication over subtraction

The distributive laws also work with subtraction. For any three integers a, b, and c

a(b−c)=ab−ac

and

(a−b)c=ac−bc

It’s not hard to show how these follow from the laws for addition. If you want to do a rigor-
ous job, the process is rather long. Table 5-3 breaks the derivation down into an S/R process,
showing every logical step, for the left-hand distributive law of multiplication over subtraction.
If you want to do a proof for the right-hand distributive law of multiplication over subtraction,
consider it a bonus exercise!

The left-hand law fails with division

The left-hand distributive laws do not work for division over addition or for division over
subtraction. To see why, all you have to do is produce examples of failure. That’s easy! Con-
sider this:

24/(4+ 2) = 24/6 = 4

but

24/4+ 24/2 = 6 + 12 = 18

And this:

24/(4− 2) = 24/2 = 12

The Distributive Laws 79

Table 5-3. Derivation of the left-hand distributive law for multiplication over

subtraction. As you read down, each statement is equal to all the statements above it.

Warning: Don’t mistake the expression (–1) for the subtraction of 1!

The parentheses emphasize that –1 is a factor in a product.

Statements Reasons

a(b−c) Begin here
a[b+ (−c)] Convert the subtraction to the addition of a negative
ab+a(−c) Left-hand distributive law of multiplication over addition
ab+ac(−1) Principle of the sign-changing element
ab+a(−1)c Commutative law for multiplication
ab+ (−1)ac Commutative law for multiplication (again)
ab+ (−ac) Principle of sign-changing element (the other way around)
ab−ac Convert the addition of a negative to a subtraction
Q.E.D. Mission accomplished