Algebra Know-It-ALL

but

24/4− 24/2 = 6 − 12 =− 6

The right-hand law works with division

If you have a sum or difference as the dividend and the single number as the divisor, you can

use the distributive law for division over addition or division over subtraction. If a and b are

any integers, and if c is any nonzero integer, then

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

and

(a−b)/c=a/c−b/c

Are you confused?

To help get rid of possible confusion about how the distributive laws operate when integers are negative,

try an example where a=−2,b=−3, and c=−4. First, work out the expression where you multiply

a times (b+c):

− 2 × [− 3 + (−4)]=− 2 × (−7)= 14

Now work out the expression where you add ab and ac:

[− 2 × (−3)]+ [− 2 × (−4)]= 6 + 8 = 14

Here’s a challenge!

With the aid of the commutative and distributive laws, prove that for any four integers a,b, c, and d

(a+b)(c+d)=ac+ad+bc+bd

Solution

Let’s do this as a narrative. Even though S/R proofs look neat, narrative proofs are often preferred by

mathematicians. We’ll start with the left-hand side of the above equation:

(a+b)(c+d)

Let’s think of the sum a+b as a single unit, and call it e. Now we can rewrite this as

e(c+d)

The left-hand distributive law for multiplication over addition allows us to rewrite this as

ec+ed

80 Multiplication and Division