Notesale: Turn your study into money

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We will start off this chapter by looking at integer exponents
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We will look at zero and negative exponents in a bit
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If a is any
number and n is a positive integer then,

So, for example,

We should also use this opportunity to remind ourselves about parenthesis and conventions
that we have in regards to exponentiation and parenthesis
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Consider the following two cases
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When performing exponentiation
remember that it is only the quantity that is immediately to the left of the exponent that gets
the power
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So, in this case we get,

In the second case however, the 2 is immediately to the left of the exponent and so it is only
the 2 that gets the power
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In this case we have the following,

We put in some extra parenthesis to help illustrate this case
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They are
important and ignoring parenthesis or putting in a set of parenthesis where they don’t belong
can completely change the answer to a problem
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Also, this warning about
parenthesis is not just intended for exponents
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Now, let’s take care of zero exponents and negative integer exponents
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This is important since
defined
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is not

We have the following definition for negative exponents
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Polynomials will show up in pretty much
every section of every chapter in the remainder of this material and so it is important that you
understand them
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Polynomials in one variable are
algebraic expressions that consist of terms in the form
where n is a non-negative
(i
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positive or zero) integer and a is a real number and is called the coefficient of the
term
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Note that we will often drop the “in one variable” part and just say polynomial
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So, a polynomial doesn’t have to contain all powers of x as we see in the first example
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We should probably discuss the final example a little more
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Remember that a polynomial is any algebraic expression that
consists of terms in the form

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Here are some examples of things that aren’t polynomials
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To see why the second one isn’t a polynomial let’s rewrite it a little
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All the exponents in the algebraic expression must be non-negative integers in
order for the algebraic expression to be a polynomial
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Let’s also rewrite the third one to see why it isn’t a polynomial
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Another rule of thumb is if there are any variables in the denominator of a fraction
then the algebraic expression isn’t a polynomial
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They
just can’t involve the variables
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Each x in the algebraic expression appears in the numerator and the exponent is a
positive (or zero) integer
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Next, let’s take a quick look at polynomials in two variables
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The degree
of each term in a polynomial in two variables is the sum of the exponents in each term and
the degree of the polynomial is the largest such sum
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In these kinds of polynomials not every term needs to have both x’s and y’s in them, in fact as
we see in the last example they don’t need to have any terms that contain both x’s
and y’s
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Note as well that multiple terms may have the same degree
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The vast majority of the polynomials that we’ll see in this course are
polynomials in one variable and so most of the examples in the remainder of this section will
be polynomials in one variable
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A monomial is a polynomial that
consists of exactly one term
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Finally, atrinomial is a polynomial that consists of exactly three terms
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Before actually starting this discussion we need to recall the distributive law
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Here is the distributive law

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