Lesson Plan
Lecture Version
Introduction to Polynomials
Objectives:
Students will:
 Analyze the graphs of polynomial functions.
 Learn vocabulary associated with polynomials.
 Determine the maximum number of real roots possible for a polynomial.
Prerequisite Knowledge
Students are able to:
 Evaluate functions for a given point.
 Graph linear and quadratic functions.
Resources
 This lesson assumes that your classroom has only one computer from
which you can lecture. For classrooms or labs with enough computers for
all students, see the lab version of this lesson.
 Rulers, 5 x 8 index cards, scissors, and paper
 Access to http://www.explorelearning.com/
 Copies of the worksheet for each students (optional)
Lesson Preparation
Before conducting this lesson, be sure to read through it thoroughly, and
familiarize yourself with the 4th degree polynomials activity at
ExploreLearning.com. You may want to bookmark the activity page for your
students. If you like, make copies of the worksheet for each student.
Lesson
Motivation
Handout an index card, a pair of scissors, and a ruler to each student. Instruct
the students to construct a box by cutting congruent squares from each corner of
the index card, then folding up the sides. The object is to construct a box with
maximum volume.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 1 of 9
Once all the boxes are constructed, have the students find the volume of their
boxes using a ruler to measure the dimensions.
Make a table listing the height and volume of each box.
Height
0
0.5
1
1.5
Volume
0
14
18
15
2
2.5
8
0
Ask the students what conjectures they can make by looking at the table. They
should be able to see that the volume varies with the height.
Add the data points from the table pictured above to the data gathered by the
students; then graph. When the points are connected, the graph should be
similar to the one below.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 2 of 9
20
18
16
14
12
10
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
Have the students use the graph to approximate the maximum volume possible
for the box. Ask the students if they know a function with this general shape.
Point out it is not a quadratic function because it does not have a vertical line of
symmetry.
Now have the students find the volume of the box algebraically in terms of x,
where x is the height of the box.
8 – 2x
5 – 2x
x by x
They should find that the volume is v(x) = 4x3 – 26x2 + 40x.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 3 of 9
Explain to the students that the expression 4x3 – 26x2 + 40x is a type of
polynomial. More precisely, it is a cubic expression because it has a degree of 3.
Tell them they will learn more about polynomials in the following activity.
The introduction to polynomials activity
To learn more about polynomials go to the 4th degree polynomials activity at
ExploreLearning.com.
Elements of polynomials
Have the students look at the graph and the equations on the screen when the
activity loads up.
Explain to the students that the degree of the polynomial is equivalent to the
highest power of the polynomial. In this case the degree is 4. Point out that the
leading coefficient is the coefficient of the highest degree term. In this case the
leading coefficient is 0.1. The constant coefficient is the term that contains no
variable term. 2 is the constant coefficient in this example.
Classes of polynomials
Polynomials are divided into two classes odd and even. Even polynomials are
polynomials that have an even degree like 3x2 + 3 or –6x42 + 2x – 8. Odd
polynomials are polynomials that have an odd degree like 8x3 + 3 or 5x31 +12x – 8.
Right and left hand behavior of even polynomials
Graph y = 2x4 – 5x3 – 4x2 + x + 8.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 4 of 9
Ask the students what happens to the graph as |x| gets large. Use the zoom tool
to zoom out. Students should see that the graph tends toward positive infinity as
|x| gets large.
Now repeat this exercise with y = x2 + 3x - 8. Have students make conjectures
about the end behavior of polynomials with a positive leading coefficient and
even degree.
Next graph y = - 2x4 – 5x3 – 4x2 + x + 8 and y = -x2 + 3x - 8.
Ask them how the negative leading coefficient changed the behavior of the
graph. Have them make conjectures about the end behavior of polynomials with
a negative leading coefficient and even degree. Students should see that the
graph tends towards negative infinity as |x| gets large.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 5 of 9
Right and left hand behavior of odd polynomials
Graph y = 2x3 + x2 – x + 1 (a=0).
Ask the students what happens to the graph as |x| gets large. Use the zoom tool
to zoom out. Students should see that the graph tends towards positive infinity as
x moves to the right and the graph tends towards negative infinity as x moves to
the left.
Now repeat this exercise with y = x + 2. Have students make conjectures about
the end behavior of polynomials with a positive leading coefficient and odd
degree.
Next graph y = -2x3 + x2 – x + 1 and y = -x + 2.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 6 of 9
Ask them how the negative leading coefficient changed to end behavior of the
graph. Have them make conjectures about the end behavior of polynomials with
a negative leading coefficient and odd degree. Students should see that the
graph tends towards negative infinity as x moves to the right and the graph tends
towards positive infinity as x moves to the left.
Have the students create a chart like the one below for future reference.
End Behavior of Polynomials
Even
Odd
Degree
Degree
Positive
Leading
Coefficient
Negative
Leading
Coefficient
Determining the maximum number of real roots
Explain to the students that finding the roots of a polynomial equation is
equivalent to finding the x-intercepts on the graph of the polynomial.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 7 of 9
Graph y = - 2x4 + 3x3 – x2 - x + 5
Ask the students how many real roots the function has. They should be able to
see this function has two x-intercepts; therefore it has two real roots. Experiment
with the slide bars of each variable. Try to find the maximum number of real roots
possible for a 4th power polynomial.
After several minutes ask the students how many real roots are possible for a 4th
power polynomial. You should have found some 4th degree polynomials with 4
real roots. An example is shown below.
Repeat this exercise with polynomials of 3rd degree, 2nd degree, and 1st degree.
Have them make conjectures relating the degree of a polynomial to the maximum
number of real roots possible for that polynomial.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 8 of 9
Back to the boxes
The volume function of the boxes was v(x) = 4x3 – 26x2 + 40x. The graph is
pictured below.
Ask the students why there are roots at 0 and 2.5. Ask the students why the
maximum volume for the box was only 18 cubic inches, when v(x) tends towards
positive infinity as x approaches positive infinity.
Conclusion
Polynomials are formed when two or more monomials are added together. The
highest power of the polynomial is the degree of the polynomial. The coefficient
of the highest-powered term is called the leading coefficient. Polynomials can be
classified as odd or even depending on the degree. The end behavior of the
graph of a polynomial is determined by the degree and the leading coefficient of
the polynomial. The maximum possible number of real roots of a polynomial is
equal to its degree. In subsequent activities you will learn more about polynomial
functions.
ExploreLearning.com Lesson Plan >> Introduction to Polynomials (Lecture Version) >> Page 9 of 9