Quite likely the greatest source of errors in Intermediate Algebra is the **Distributive Law**

There are several subtleties in this law, any of which can give rise to errors.

You should work through all of the following items and make sure you understand what they say. If it helps assign values to the variables, such as

- There is nothing magical about the letters , and
.
It's the

**pattern**that matters: you**multiply a sum with a factor by multiplying each term in the sum with that factor.** - A more general version of the Distributive Law is that the
product of two sums is the sum of all possible products of one term
from the first sum and one term for the second sum. For example
- The variables , and may be algebraic
expressions. For example, might equal in which case the
Distributive Law becomes
**FOIL**(first-outer-inner-last) but that's just a special case of the Distributive Law. It does not pay to commit such special cases to memory. - Since division is equivalent to multiplication with the
reciprocal, division is just a special case of multiplication.
Hence a
formula such as
- It is tempting to omit the parentheses but
in general
- Similarly, when canceling common factors in a ratio one has
to make sure that the factor is in fact a factor in each term of
both the
numerator and denominator.
For example
- A common fallacy is to think that you square a sum by squaring
each term. You don't!
- A special case of , and being algebraic
expressions is that they have a negative sign, in which case we get
formulas like
- The Distributive Law works in both directions, so you can use it
to
*factor out*a term. - One of the factors may be and not always visible, for
example