6.2: Polynomials
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- Apr 17, 2021
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We begin with the definition of a term.
Definition: Term
A term is either a single number (called a constant term) or the product of a number and one or more variables.
For example, each of the following is a term.
Note how the first term is a single number, while the remaining terms are products of a number and one or more variables. For example,
Definition: Coefficient
When a term is a product of a number and one or more variables, the number is called the coefficient of the term. In the case of a term that is a single number, the number itself is called the coefficient.
Thus, for example, the coefficients of the terms
Definition: Degree
The degree of a term is the sum of the exponents on each variable of the term. A constant term (single number with no variables) has degree zero.
Thus, for example, the degrees of the terms
Definition: Monomial
The words monomial and term are equivalent.
Thus,
Definition: Binomial
A binomial is a mathematical expression containing exactly two terms, separated by plus or minus signs.
For example, each of the mathematical expressions
Definition: Trinomial
A trinomial is a mathematical expression containing exactly three terms, separated by plus or minus signs.
For example, each of the mathematical expressions
A bicycle has two wheels, a binomial has two terms. A tricycle has three wheels, a trinomial has three terms. But once we get past three terms, the assignment of special names ceases and we use the generic word polynomial, which means “many terms.”
Definition: Polynomial
A polynomial is a many-termed mathematical expression, with terms separated by plus or minus signs. The coefficients of a polynomial are the coefficients of its terms.
Each of the previous expressions,
Ascending and Descending Powers
When asked to simplify a polynomial expression, we should combine any like terms we find, and when possible, arrange the answer in ascending or descending powers.
Example
Simplify the following polynomial expression, arranging your answer in the descending powers of
Solution
In order to arrange our answer in descending powers of
Note how the powers of
To arrange our final answer in ascending powers of
Note how we start with the constant term, then the powers of
Exercise
Simplify the following polynomial, and arrange your answer in ascending powers of
- Answer
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When we have a polynomial in a single variable, such as the polynomial in Example
Example
Simplify the following polynomial expression, then arrange your answer in descending powers of
Solution
We’ll again use the commutative and associative properties to change the order and regroup, putting the terms with the highest powers of
Note that this is a very natural order, the powers of
Exercise
Simplify the following polynomial, and arrange your answer in descending powers of
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Not all examples will have nice ordering presented in Example
Example
Simplify the following polynomial expression, then arrange your answer in some sort of reasonable order.
Solution
Let’s try to arrange the terms so that the powers of a descend. Again, we use the commutative and associative properties to change the order and regroup.
Note that in our final arrangement, the powers of
Exercise
Simplify the following polynomial, and arrange your answer in ascending powers of
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The Degree of a Polynomial
To find the degree of a polynomial, locate the term of the polynomial having the highest degree.
The degree of a polynomial
The degree of a polynomial is the degree of the term having the highest degree.
Finding the degree of a polynomial of a single variable is pretty easy.
Example
What is the degree of the polynomial
Solution
First, let’s arrange the polynomial in descending powers of x.
Arranging the polynomial in descending powers of
Exercise
What is the degree of the polynomial
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Finding the degree of a polynomial of more than one variable is a little bit trickier.
Example
What is the degree of the polynomial
Solution
Note that the polynomial is already arranged in descending powers of
Hence, the term with the highest degree is
Exercise
What is the degree of the polynomial
- Answer
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Polynomial Functions
First we define what we mean by a polynomial function.
Polynomial function
A polynomial function is a function defined by a rule that assigns to each domain object a range object defined by a polynomial expression.
Advanced courses, such as multivariate calculus, frequently use polynomial functions of more than one variable such as
Example
Given the polynomial function
Solution
To evaluate
Next, substitute
Hence,

Exercise
Given the polynomial function
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The Graph of a Polynomial Function
One of the most important polynomial functions in all of mathematics and science is the polynomial having degree two.
Quadratic polynomial
The second degree polynomial having the form
The parabola is approximately U-shaped. Some open upwards, some open downwards, depending on the sign of the leading term.
In Figure

In Figure

Note
The sign of the leading term of
- If
, the parabola opens upward. - If
, the parabola opens downward.
The turning point of a parabola has a special name.
The vertex of a parabola
The graph of the second degree polynomial
Example
Use your graphing calculator to sketch the graph of the quadratic polynomial
Solution
The degree of the polynomial

Note that the graph in Figure

In reporting your result on your homework, follow the Calculator Submission Guidelines from Chapter 3, Section2.
- Draw axes with a ruler.
- Label the horizontal axis
and the vertical axis . - Indicate the WINDOW parameters
, and \) at the end of each axis. - Freehand the curve and label it with its equation.
Exercise
Use your graphing calculator to sketch the graph of the quadratic polynomial
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When the degree of the polynomial is larger than two, the number of turning points of the graph might increase. This makes for some very interesting curves. In more advanced courses, such as intermediate and college algebra, you will be introduced to a variety of techniques that will help you determine appropriate viewing windows for the graphs of these higher degree polynomials. However, in this introductory section, we will assist you by suggesting a good viewing window for each polynomial, one that will allow you to see all of the turning points of the graph of the polynomial.
Example
Use your graphing calculator to sketch the graph of the polynomial function
Solution
Enter the polynomial function in

Push the GRAPH button on the top row of your calculator to produce the graph of the polynomial function shown in Figure

Sweet-looking curve!
Exercise
Use your graphing calculator to sketch the graph of the quadratic polynomial p(x)=x3 −14x2 + 20x+ 60. Set your window parameters as follows:
- Answer
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