Download Basic Algebraic Operations and Simplification

The Mathematics 11
Competency Test
Basic Algebraic Operations and
Simplification
For the purposes of the BCIT Mathematics 11 Competency Test, 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.
David W. Sabo (2003)
Basic Algebraic Operations and Simplification
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