The distributive law describes how multiplication distributes over addition and subtraction.
I want to start by showing you how it works using numbers first, so you can get your head
around what’s happening and why it works. Have a look at this example. Three, times
one-plus-four. That’s three lots, of one plus four. Well, one plus four makes five
in each row. So I have three times five, which is fifteen.
But I can also look at the same question a different way. I have three lots of one, and
three lots of four. Three times one, plus three times four. Three, plus twelve, is fifteen.
See? I got the same answer, but I worked it out two different ways. What we’re going
to do is apply this idea to algebraic variables as well as numbers. Because sometimes I want
a product, but other times I want a sum of terms instead. So I want to be able to convert
back and forth between the two forms.
Suppose instead of four, I have a. I don’t know how much a is worth; it’s just a. So
I can’t actually add one to a yet; I can’t evaluate one plus a without a value for a.
But that’s okay; I can still expand it using the distributive law. Because I have three
lots of one, plus three lots of a. But three times one is just three, and three times a
we normally write as just three a.
See how that worked? I applied the three out the front of the brackets to each of the terms
inside the brackets. And I turned my product into a sum.
Let’s try that again. Four brackets b-plus-six. Remember there’s a multiplication in there,
between the four and the bracket: four, times b-plus-six. Now I don’t know how big b is;
I don’t have a value for it. But I can still expand it using the distributive law. I have
four lots of b, plus four lots of six. Four b, plus twenty-four.
See how easy that was? I turned my product, four times the brackets, into a sum, four
b plus twenty-four.
When we start adding more variables, drawing the diagrams becomes a bit more difficult.
So from now on, we’re going to do it just using the algebraic process. To expand a product
using the distributive law, rewrite the expression without the brackets, by distributing the
multiplication over the addition or subtraction.
Two brackets c plus three. Remember there’s a hidden times in there. Distribute the two
over the terms in brackets. Two times c, plus two times three. Which we can write as two
c plus six.
When you get better at this, you are allowed to skip the second line here. You don’t
have to write the extra step. But I’m going to keep doing it for now, so you can see clearly
exactly what I’m doing; and you are allowed to keep writing it every time, if it helps
you to get the right answer.
Eight, brackets two-minus-p. Distribute the eight over what’s inside the brackets. Eight
times two, minus eight times p. The minus sign just gets copied down from inside the
brackets. Eight times two is sixteen, so it’s sixteen minus eight p.
Let’s try it with some negative numbers. Nothing will change, it’s done exactly the
same way. We just have to be careful to get the signs right.
Negative four times brackets a-plus-two. Distribute the times over the brackets. Negative four
times a, plus negative four times two. That’s negative four a, plus negative eight.
Negative g, times brackets three-minus-f. Watch those minus signs. Distribute it over
the brackets. Negative g times three, minus, negative g times f. Can you see where each
of those negative and minus signs went? Be very careful now; don’t lose any of those
signs. It’s negative three g, minus negative f g; I’ve just written the same thing again
without the times signs, and put the g and the f in the second term into alphabetic order.
Now this answer is correct. But it’s not quite as simple as it could be written. I
don’t know about you, but I don’t really like subtracting a negative. I’d rather
add a positive instead. Subtracting a negative is the same as adding a positive. So I can
write it as negative three g, plus f g.
Now, from here we just need some more practice. Let me show you a couple more examples of
what can happen.
q, times brackets four q plus y squared. Exactly as before, distribute the times over the sum
in brackets. q times four q, plus q times y squared. Now simplify the first term using
power notation: four q squared, plus q y squared. (Remember q y squared means q, times y squared,
not q-y, all squared. That’s order of operations again: exponents happen first, before the
multiplication.)
What if the brackets are on the left instead? n plus two v, in brackets, times six v. Well,
it’s still a product, so I can still distribute the factor on the right over the addition
in brackets on the left. n times six v, plus two v times six v. See? It’s exactly the
same. Each term in the brackets has been multiplied by the factor that was outside the brackets.
So now just simplify these terms: six n v, in alphabetic order, plus twelve (two times
six) v squared. Easy.