Distributivity

Now that we understand parentheses and order, we can learn a very important rule for distributing a product or factoring a sum called the rule of distributivity. The following rule is used in either direction - left to right to distribute, right to left to factor:

$$a(b+c) = ab + ac \quad {\rm distributivity}$$ (3.27)

Note well that we are no longer writing in an explicit $*$. Did you read this (correctly) as:

$$a*(b+c) = (a*b) + (a*c)$$ (3.28)

where we used the conventions defined above to determine the implicit order and sign?

Our repertoire of algebraic transformations is almost complete, or at least almost complete enough that we can solve a whole lot of things with them as is or use them to quickly derive new rules below that aren't worth putting in our essential list even though they may well be very useful for certain problems.