December 19th
December 19th
Basic Algebraic Operations and Simplification
We can say that the “basic” algebraic operations you
must master are:
 addition (of two or more terms or expressions)
 subtraction (of one term or expression from another)
 multiplication (of one term or expression by another) In
the case of multiplication, it turns out to be useful not
only to start with two expressions and be able to write
down the result of multiplying them together, but also,
to be able to start with an expression and rewrite it as
the product of two or more factors.
 division (of one term or expression by another –
this leads to the whole subject of working with fractions
containing literal symbols.)
 manipulating radicals or roots, particularly square
roots, of algebraic expressions
Because the presence of literal symbols in algebraic
expressions can lead to considerable (indeed terrifying)
complexity when some of these basic operations are performed, a
very important algebraic skill is the ability to simplify
algebraic expressions of various sorts, whenever such a thing is
possible. Simplification is something you do a lot in algebra
(and in mathematics in general), though it is a bit difficult to
define precisely what is meant by one expression being simpler
than another in all situations. Also, exactly how one might
achieve a simplification depends on the features of the algebraic
expression with which you are dealing. In general:
one expression is simpler than another if it has fewer
terms, or if its parts have fewer terms (for example, in the case
of fractions)
The catch is that in the process of simplifying an expression,
we must make sure that the new simpler expression is
mathematically equivalent to the original expression – it
evaluates to the same value as the original expression whenever
the same values are substituted for corresponding literal symbols
in the two. The rules and strategies for simplification that we
will describe are intended to ensure that this requirement is
satisfied. By imposing this requirement, we are able to discard
the original more complicated expressions and continue to work
with the simplified version since in the end the simplified
version must give exactly the same results.
When doing algebra, it is generally expected that where an
“obvious” simplification of an expression is possible,
you will carry out that simplification before stating a final
solution to a problem. What are “ obvious ” possible
simplifications to check for depends on the situation. The most
important and common strategies for simplification of various
types of expressions will be described with examples in the next
few sections of these notes.
