Identify polynomial functions

We have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions. Polynomials are algebraic expressions that are created by summing monomial terms, such as [latex]-3x^2[/latex], where the exponents are only integers. Functions are a specific type of relation in which each input value has one and only one output value. Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section we will identify and evaluate polynomial functions. Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable.

When we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[/latex].  We can turn this into a polynomial function by using function notation:

[latex]f(x)=4x^3-9x^2+6x[/latex]

Polynomial functions are written with the leading term first, and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.

In the following video you will see additional examples of how to identify a polynomial function using the definition.

Define the degree and leading coefficient of a polynomial function

Just as we identified the degree of a polynomial, we can identify the degree of a polynomial function.  To review: the degree of the polynomial is the highest power of the variable that occurs in the polynomial; the leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

In the next video we will show more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.

Graphs of Polynomial Functions

Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes impossible to find by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.

Polynomial functions of degree 2 or more have graphs that do not have sharp corners these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.

Now you try it.

Identifying the shape of the graph of a polynomial function

Knowing the degree of a polynomial function is useful in helping us predict what it’s graph will look like. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the graph of the polynomial will match the end behavior of the term of highest degree.

As an example we compare the outputs of a degree 2 polynomial and a degree 5 polynomial in the following table.

As the inputs for both functions get larger, the degree 5 polynomial outputs get much larger than the degree 2 polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of polynomial graphs.

There are two other important features of polynomials that influence the shape of it’s graph. The first  is whether the degree is even or odd, and the second is whether the leading term is negative.

Even degree polynomials

In the figure below, we show the graphs of [latex]f\left(x\right)={x}^{2},g\left(x\right)={x}^{4}[/latex] and [latex]\text{and}h\left(x\right)={x}^{6}[/latex], which are all have even degrees. Notice that these graphs have similar shapes, very much like that of a quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

Odd degree polynomials

The next figure shows the graphs of [latex]f\left(x\right)={x}^{3},g\left(x\right)={x}^{5},\text{and}h\left(x\right)={x}^{7}[/latex], which are all odd degree functions.

Notice that one arm of the graph points down and the other points up.  This is because when your input is negative, you will get a negative output if the degree is odd. The following table of values shows this.

Now you try it.

Example

Identify whether graph represents a polynomial function that has a degree that is even or odd.

a)

b)

Show Answer

a) Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even.  If you apply negative inputs to an even degree polynomial you will get positive outputs back.

b) As the inputs of this polynomial become more negative the outputs also become negative, the only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have odd degree.

Put Answer Here

 The sign of the leading term

What would happen if we change the sign of the leading term of an even degree polynomial?  For example, let’s say that the leading term of a polynomial is [latex]-3x^4[/latex].  We will use a table of values to compare the outputs for a polynomial with leading term [latex]-3x^4[/latex], and [latex]3x^4[/latex].

Plotting these points on a grid leads to this plot, the red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:

The negative sign creates a reflection of [latex]3x^4[/latex] across the x-axis.  The arms of a polynomial with a leading term of [latex]-3x^4[/latex] will point down, whereas the arms of a polynomial with leading term [latex]3x^4[/latex] will point up.

Now you try it.

Example

Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.

a)

b)

Show Answer

a) Both arms of this polynomial point in the same direction so it must have an even degree.  The leading term of the polynomial must be negative since the arms are pointing downward.

b) The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.

Just as we have performed algebraic operations on polynomials, we can do the same with polynomial functions. In this section we will show you how to perform algebraic operations on polynomial functions and introduce related notation.

The four basic operations on functions are adding, subtracting, multiplying, and dividing. The notation for these functions is as follows.

Addition [latex](f + g)(x) = f(x)+ g(x)[/latex]

Subtraction [latex](f − g)(x)= f(x) − g(x)[/latex]

Multiplication [latex](f · g)(x)= f(x)g(x)[/latex]

Division  [latex]\frac{f}{ g} (x) = \frac{f(x)}{g(x)}[/latex]

We will focus on applying these operations to polynomial functions in this section.

Add and subtract polynomial functions

Adding and subtracting polynomial functions is the same as adding and subtracting polynomials. When you evaluate a sum or difference of functions, you can either evaluate first, or perform the operation on the functions first, as we will see. Our next examples describe the notation used to add and subtract polynomial functions.

In our next example we will evaluate a sum and difference of functions and show that you can get to the same result in one of two ways.

Multiply and divide polynomial functions

We saw that multiplying polynomials often required the use of the distributive property, and that the algebra of dividing polynomials could get messy fast!  The same techniques can be used to multiply and divide polynomial functions. Additionally, the same idea applies to evaluating a product or quotient of functions as we discovered in the previous example. We can either evaluate the function and then perform the indicated operation, or vice-versa. You may already be thinking – it will be a lot less work to evaluate the polynomials and then divide the results!

In the next example we will divide polynomial functions and then evaluate the new function.

Example

Given [latex]p(x)=2x^2+x-15[/latex] and [latex]r(x)=x+3[/latex]

Find [latex]\frac{p}{r}(x)[/latex] and evaluate [latex]\frac{p}{r}(2)[/latex]

Show Answer

We can use synthetic division for this polynomial division since the coefficient on [latex]r(x)=x+3[/latex] = 1.

This result means that [latex]\frac{p}{r}(x)=\frac{2x^2+x-15}{x+3}=2x-5[/latex]

Let’s evaluate this quotient for x = -3 both ways as we did in a previous example.

First, we will evaluate the result after polynomial division:

[latex]\begin{array}{ccc}\frac{p}{r}(x)=2x-5\\\frac{p}{r}(2)=2(2)-5=4-5=-1\end{array}[/latex]

Next, we will evaluate each function for x = 2, then we will divide the results.

[latex]p(2)=2(2)^2+(2)-15=8+2-15=-5[/latex]

[latex]r(2)=(2)+3=5[/latex]

Divide the results:

[latex]\frac{p}{r}(2)\frac{-5}{5}=1[/latex]

Summary

Polynomial functions contain powers that are non-negative integers and the coefficients are real numbers. it is often helpful to know how to identify the degree and leading coefficient of a polynomial function. To do this, follow these suggestions:

1. Find the highest power of x to determine the degree function.
2. Identify the term containing the highest power of x to find the leading term.
3. Identify the coefficient of the leading term.