We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Hướng dẫn học tập > Intermediate Algebra

8B.2 Polynomial Functions

8B.2.1 Learning Objectives

  • Introduction to polynomial functions
    • Identify polynomial functions
    • Identify the degree and leading coefficient of a polynomial function
  • Algebra of Polynomial Functions
    • Add and subtract polynomial functions
    • Multiply and divide polynomial functions

8B.2.1 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.

Example 8B.2.A

Which of the following are polynomial functions?

[latex]\begin{array}{ccc}f\left(x\right)=2{x}^{3}\cdot 3x+4\hfill \\ g\left(x\right)=-x\left({x}^{2}-4\right)\hfill \\ h\left(x\right)=5\sqrt{x}+2\hfill \end{array}[/latex]

Answer:

The first two functions are examples of polynomial functions because they contain powers that are non-negative integers and the coefficients are real numbers.

  • [latex]f\left(x\right)[/latex] can be written as [latex]f\left(x\right)=6{x}^{4}+4[/latex].
  • [latex]g\left(x\right)[/latex] can be written as [latex]g\left(x\right)=-{x}^{3}+4x[/latex].
  • [latex]h\left(x\right)=5\sqrt{x}+2[/latex] is not a polynomial because the variable is under a square root - therefore the exponent is not a positive integer.

In the following video you will see additional examples of how to identify a polynomial function using the definition. https://youtu.be/w02qTLrJYiQ  

8B.2.2 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.

Example 8B.2.B

Identify the degree, leading term, and leading coefficient of the following polynomial functions.
[latex]\begin{array}{ccc} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\ g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\ h\left(p\right)=6p-{p}^{3}-2\end{array}[/latex]

Answer:

For the function [latex]f\left(x\right)[/latex], the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[/latex]. The leading coefficient is the coefficient of that term, –4.

For the function [latex]g\left(t\right)[/latex], the highest power of t is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[/latex]. The leading coefficient is the coefficient of that term, 5.

For the function [latex]h\left(p\right)[/latex], the highest power of p is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[/latex]; the leading coefficient is the coefficient of that term, –1.

In the next video we will show more examples of how to identify the degree, leading term and leading coefficient of a polynomial function. https://youtu.be/F_G_w82s0QA

8B.2.3 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. Graph of f(x)=x^3-0.01x. Now you try it.

Example 8B.2.C

Which of the graphs below represents a polynomial function? Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.

Answer:

The graphs of f and h are graphs of polynomial functions. They are smooth and continuous.

The graphs of g and k are graphs of functions that are not polynomials. The graph of function g has a sharp corner. The graph of function k is not continuous.

Q & A

Do all polynomial functions have as their domain all real numbers?

Yes. Any real number is a valid input for a polynomial function.

8B.2.4 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.
x [latex]f(x)=2x^2-2x+4[/latex] [latex]g(x)=x^5+2x^3-12x+3[/latex]
1 4 8
10 184 98117
100 19804 9998001197
1000 1998004 9999980000000000
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.

Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.

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. Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.

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.

x [latex]f(x)=x^4[/latex] [latex]h(x)=x^5[/latex]
-1 1 -1
-2 16 -32
-3 81 -243
Now you try it.

Example 8B.2.D

Identify whether graph represents a polynomial function that has a degree that is even or odd. a) Graph of f(x)=5x^4+2x^3-x-4. b) Graph of f(x)=3x^5-4x^4+2x^2+1.

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].
x [latex]-3x^4[/latex] [latex]3x^4[/latex]
-2 -48 48
-1 -3 3
0 0 0
1 -3 3
2 -48 48
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: Screen Shot 2016-07-15 at 2.21.36 PM 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 8B.2.E

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) Graph of f(x)=-2x^6-x^5+3x^4+x^3. b) Graph of f(x)=-6x^3+7x^2+3x+1.

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.

8B.2.2 Learning Objectives

  • Add and subtract polynomial functions
  • Multiply and divide polynomial functions
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.

8B.2.5 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.

Example 8B.2.F

For [latex]f(x)=2x^3-5x+3[/latex] and [latex]h(x)=x-5[/latex], Find the following: [latex-display](f+h)(x)[/latex] and [latex](h-f)(x)[/latex-display]

Answer: [latex-display]\begin{array}{ccc}(f+h)(x)=f(x)+ h(x)=(2x^3-5x+3)+(x-5)\\=2x^3-5x+3+x-5\,\,\,\,\,\text{combine like terms}\\=2x^3-4x-2\,\,\,\,\,\text{simplify}\end{array}[/latex-display] [latex-display]\begin{array}{ccc}(h-f)(x)=h(x)-f(x)=(x-5)-(2x^3-5x+3)\\=x-5-2x^3+5x-3\,\,\,\,\,\,\text{combine like terms}\\=-2x^3+6x-8\,\,\,\,\,\text{simplify}\end{array}[/latex-display]

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.

Example

For [latex]f(x)=2x^3-5x+3[/latex] and [latex]h(x)=x-5[/latex] Evaluate: [latex](f+h)(2)[/latex] Show that you get the same result by 1)evaluating the functions first, then performing the indicated operation on the result and 2) performing the operation on the functions first, then evaluating the result

Answer:   1)[latex](f+h)(2)[/latex] First, we will evaluate the functions separately: [latex-display]f(2)=2(2)^3-5(2)+3=16-10+3=9[/latex-display] [latex-display]h(2)=(2)-5=-3[/latex-display] Now we will perform the indicated operation using the results: [latex-display](f+h)(2)=f(2)+h(2)=9+(-3)=6[/latex-display] 2) We can get the same result by adding the functions first, then evaluating the result at x=2 [latex](f+h)(x)=f(x)+h(x)=2x^3-4x-2[/latex] from above. Now we can evaluate this result at x=2 [latex-display](f+h)(2)=2(2)^3-4(2)-2=16-8-2=6[/latex-display] Both methods give the same result, and both require about the same amount of work.

 

8B.2.6 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!

Example 8B.2.G

Given: [latex]g(t)=2t^3-t^2+7[/latex] and [latex]f(t)=5t^2-3[/latex] Find: [latex](g · f)(t)[/latex], and evaluate [latex](g · f)(-1)[/latex]

Answer: [latex-display]\begin{array}{ccc}(g · f)(t)=(2t^3-t^2+7)(5t^2-3)\\=(2t^3\cdot(5t^2)-t^2\cdot(5t^2)+7\cdot(5t^2))+(2t^3\cdot(-3)-t^2\cdot(-3)+7\cdot(-3))\,\,\,\,\,\text{apply the distributive property}\\=(10t^5-5t^4+35t^2)+(-6t^3+3t^2-21)\,\,\,\,\text{simplify}\\=10t^5-5t^4-6t^3+38t^2-21\,\,\,\,\,\text{combine like terms}\end{array}[/latex-display] Evaluate [latex](g · f)(-1)[/latex] [latex-display]\begin{array}{ccc}(g · f)(t)=10t^5-5t^4-6t^3+38t^2-21\\(g · f)(-1)=10(-1)^5-5(-1)^4-6(-1)^3+38(-1)^2-21\\=-10-5+6+38-21\\=8\end{array}[/latex-display]

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

Example 8B.2.H

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]

Answer: We can use synthetic division for this polynomial division since the coefficient on [latex]r(x)=x+3[/latex] = 1. Screen Shot 2016-07-15 at 4.09.44 PM 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 = 2 both ways as we did in a previous example. First, we will evaluate the result after polynomial division: [latex-display]\begin{array}{ccc}\frac{p}{r}(x)=2x-5\\\frac{p}{r}(2)=2(2)-5=4-5=-1\end{array}[/latex-display] Next, we will evaluate each function for x = 2, then we will divide the results. [latex-display]p(2)=2(2)^2+(2)-15=8+2-15=-5[/latex-display] [latex-display]r(2)=(2)+3=5[/latex-display] Divide the results: [latex-display]\frac{p}{r}(2)=\frac{-5}{5}=-1[/latex-display]

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.
 

Licenses & Attributions

CC licensed content, Original

  • Determine if a Function is a Polynomial Function. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Degree, Leading Term, and Leading Coefficient of a Polynomial Function. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.

CC licensed content, Shared previously