5.1 The Distributive Law
In chapter 8.4, Operators and Calculation rules, we went through the distributive law and now we will work with algebraic expression. The distributive law connects multiplication with addition and subtraction.  We usually call it for “multiplying in” or “factoring out”.


Multiplying in

You know that 3 · 9 = 27

We can write 9 as a sum of two terms, for example (5 + 4).

If we multiply 3 with (5 + 4), will we get the same results as if we had multiplied 3 by 9?

We illustrate this in pictures:
Here you see that we will get the same result, 27. You can try to write 9 as a sum of other terms as see that you will get the same result.

If we change out the number for variables: 3 with a, 5 with b and 4 with c, the expression looks like this:

a(b + c) = ab + ac


In your mathematics book, it may have been explained like this:
x(y + z) = (x · y) + (x · z) = xy + xz

Factorising out

Example A
We have two terms and will take out the largest common factor.

15 + 12 have a common factor, 3. 3 is the then the largest common factor for 15 and 12.

We rewrite the terms as products.

15 is written as (3 · 5) and 2 is written as (3 · 4). Then we can write the expression like this:
15
+
12
(3 · 5)
+
(3 · 4)
We take out the common factor 3.

We have now factorised the expression.


Example B

We will take an algebraic example.

15t3 + 12t2u - 6tu2

The largest common factor is 3t.

We rewrite the terms as products and take out the 3t.

(3t · 5t2) + (3t · 4tu) - (3t · 2u2) = 3t (5t2 + 4tu - 2u2)