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7.1 - Properties of Exponents

7.1 - Properties of Exponents

Author: Kyle Webb

In this unit, we will work with higher order polynomials.  We have explored the classes of linear functions and quadratic functions in our earlier work, and this unit will build on your understanding of the behavior of functions to introduce third-degree and higher-degree polynomials.  Before we begin this work, we must spend some time reviewing our understanding of polynomials and working with the properties of exponents. 

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What are Polynomials?

A polynomial is a mathematical expression that consists of variables and constants separated into terms by the operations of addition, subtraction, and multiplication, and with only non-negative integer exponents.  A polynomial can be classified by the number of terms it has as well as by the degree of the polynomial.  The table below gives the names of polynomials as classified by the number of terms.

A polynomial with more than three terms is classified as simply a polynomial.  In addition to using the number of terms to name a polynomial, we can name a polynomial based on its degree.  Let’s first consider polynomials in one variable.  For example,

negative 2 x squared minus 3 x plus x to the power of 4 plus 7

To classify this polynomial expression by degree, we rewrite the polynomial so that the exponents are in descending order.

x to the power of 4 minus 2 x squared minus 3 x plus 7

Since the highest exponent in this polynomial is 4, we would classify this as a degree 4 polynomial. 

It becomes a little more complicated when we have a polynomial in more than one variable.  Consider,

4 x squared y to the power of 4 minus 3 x y cubed plus 2 y squared minus 5

Now we have both x and y with exponents.  To determine the degree, you must add the exponents for an individual term, and use the term with the largest sum.  In this example, we would use to conclude that the degree of this polynomial is 6.  We won’t worry too much about writing polynomials in more than one variable in a correct order, but if you are interested in knowing the accepted practice for this, choose one variable, and write the terms in the descending order for that variable.  The polynomial above is written in descending order. 

The table below shows polynomial names by degree up to degree 4.

Source: MICDS

Properties of Exponents

Rational Exponents

Properties of Exponents Graphic Organizer


Source: Kyle Webb