Polynomials: Basic Operations

Definition of positive integer exponents
In the notation , is called the base and is called the exponent
or the power.
When the exponent tells you how many times to multiply
the base by itself.
     
where you have factors in the product on the right side of the equation.
     .

In particular, when you don't see an exponent in an expression, by convention
the exponent is supposed to be ,
     .

When the exponent is the number zero,
, provided that .

is called an indeterminate expression.


Some other conventions to keep in mind are

       

    

A basic property for working with exponents is

       .

In words, when you multiply powers to the same base you add the exponents.

      


Polynomials: Terminology
A monomial is a string of numbers, symbols, and variables connected by
the operation of multiplication and any exponents that occur are positive
integers. A variable cannot appear in a denominator, as an exponent, or
inside a radical sign.

The following are examples of monomials:
                           ,   , 

A polynomial is a made up of monomials added or subtracted one after
the other.
Examples are
        is a binomial

        is a trinomial

         is a polynomial of 4 terms.


The Distributive Property
                                                        

Multiply everything inside the parenthesis by .
Remember that is a symbol that can stand for almost anything
mathematical. That is, might be a simple number like .
Or might be a complicated algebraic expression.
In any case, we multiply everything inside the parenthesis by .

Given the problem
          
probably we all would do the stuff inside the parenthesis first
            
and then multiply that by the number outside the parenthesis
       .

The distributive property says that there is another way to the problem,
namely,
      .
You get the same answer.
In arithmetic we may have a choice whether or not to use the distributive
property. But in algebra, very often we do not have a choice.
In the problem
     
we can not say that    is something the way we can say that   
is .
            remains , so we have no choice but to use the
           distributive property to perform the indicated multiplication,
      .

Change some of the numbers and some of the signs for illustration:
     
      .

The distributive property is used to do multiplications, to simplify expressions,
and to combine like terms.
For example, how would you justify the solution to the problem of combining
the like terms in
       ?
If we were talking about coconuts, you would simply count the number of
physical coconuts. But we are not dealing with coconuts here, we are
dealing with mathematical symbols. To justify combining like terms, we
need to invoke the algebraic distributive property,
     
and then do what's inside the parenthesis

                     .


See Examples 2 - 5, pages 15 - 18.

Now let's look at some very important Special Products.
Using the distributive property, multiply everything inside the
first parentheses by everything inside the second parenthesis:







Caution:



For example,





See Example 6, page 19.


Order of Operations
When we have to simplify some complicated-looking expression involving
parenthesis, powers, and so on, sometimes we wonder - where do we start,
what do we do first?
The order of operations answers that question. It is a good guide to follow.
First, do what is inside a parenthesis. If there are parenthesis inside parenthesis,
do what is inside the innermost parentheses and then work yourself out of the
various parentheses.
Second, do the powers. Work out the exponentiations.
Third, do the multiplications and divisions.
Finally, do the additions and subtractions.

See Example 7, page 20.


                   top
                   next     Polynomials:Factoring