XV.  The Distributive Property

The distributive property is one of the most important in algebra.  Just like when you distribute newspapers or candy, it means you are giving something to more than one person or thing.

In math it means that when you have a problem like 3(5x + 7) you distribute the 3 times the 5x and again times the 7.  So that would equal 15x + 21.

A string and circle diagram showing distribution.  Mostly decorative.

FOIL and The Distributive Property

Again, whatever works for you - is what you should use. I have never liked F.O.I.L., but it's bound to come up - so, I'll explain it. F.O.I.L. stands for First, Outer, Inner, Last. So, in an expression like (2x + 3)(5x + 8) - with F.O.I.L. you multiply the first times the first and you get 2x * 5x or 10x2. Because 2 * 5 = 10 and x * x = x2.

The second term is outer * outer.  Well 2x is on the left outer end and 8 is on the right outer end.  So, 2x * 8 = 16x.  Inner * inner, the inside numbers are 3 and 5x and 3 * 5x = 15x.  Then Last * last or 3 * 8 = 24.

Put it all together and you get, 10x2 + 16x + 15x + 24. Combine the like terms and your expression becomes 10x2 + 31x + 24.

What's wrong with F.O.I.L.?

Maybe it's a learning style thing? The fact is, what you are doing is multiplying each thing in the first parenthesis times everything in the second parenthesis. That's it. Now, use F.O.I.L. on this problem - (2x + 3) (3x2 + 2x + 6). See the problem? Now, let's use my method. There's a 2x in the first parenthesis, so let's multiply it times each thing in the second parenthesis. 2x * 3x2 = 6x3. 2x * 2x = 4x2. 2x * 6 = 12x. So, that's done. Now we do the same thing with the next term in the first parenthesis the 3. 3 * 3x2 = 9x2. 3 * 2x = 6x. And 3 * 6 = 18. We end up with 6x3 + 4x2 + 12x + 6x + 18, now we combine like terms and end up with 6x3 + 4x2 + 18x + 18

So, the problem with F.O.I.L. is that it only works in a few cases.  Multiply Each times every works all the time.