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.