Lesson Plans
for
Polynomial Division
Real Zeros of Polynomial
and
Complex Numbers
Objectives:
Upon completion of this lesson, the students will be able to:
Polynomial Division
 Divide polynomials using long division
 Divide polynomials using synthetic division (where appropriate)
 Use the Remainder Theorem to evaluate a polynomial
 Use the Factor Theorem to factor a polynomial
Complex Numbers
 Perform operations with complex number and write the results in standard
form
 Solve a quadratic equation involving complex zeros
Polynomials
 Find all possible rational zeros of a polynomial function using the Rational
Root Test
 Find the real zeros of a polynomial function algebraically
 Approximate the real zeros of a polynomial function using the Intermediate
Value Theorem
 Approximate the real zeros of a polynomial using a graphing utility
Prior to these sections the students should know how to evaluate a polynomial at a given
point, factor basic polynomials, and find zeros of quadratics and factorable third degree
polynomials.
Polynomial Division: (50 minutes)
Review:
Remind the students that in the early school grades they were taught how to write
numbers in expanded form ( 60785  6  10  0  10  7  10  8  10  5 ) and
how to do long division. Point out that a number written in expanded form looks very
much like a polynomial with the x replaced with 10. Polynomial long division works that
same way that long division of numbers works.
4
3
2
Also remind students of the vocabulary that is used with division (dividend, divisor,
quotient and remainder). These have the same meaning they did when they used them in
numerical division.
Material:
We start with long division:
Work through the first example explaining how each step works. The first example is a
simple, straightforward example with no remainder.
Example 1: Divide by long division.
Dividend
Divisor
 3x 2  7 x  4
x 1
Set up the second example with the students and then let then complete the problem. The
second example requires using a placeholder for a “missing” term in the divisor.
Example 2: Divide by long division.
Dividend
Divisor
4
3
2
 x  2 x  3x  8 x  4
x2  4
When going over the example, talk with the students about how to deal with the
remainder.
Give the students the third example to do on their own.
Example 3: Divide by long division.
Dividend
Divisor
3
 x 9
x2 1
Since long division takes a good deal of time. You can introduce synthetic division as a
time saver for certain types of polynomial division. It works well to introduce it as a
short hand when the divisor is of the form x  c .
Work through the fourth example explaining how each step works. The fourth example
is a simple, straightforward example.
Example 4: Divide by long division.
Dividend
Divisor
3
2
 2 x  5x  7 x  20
x4
Make sure that the students have to write out the polynomial that is the quotient (not just
the coefficients).
Ask the students how to set up the fifth example (make sure they see that placeholders are
necessary in synthetic division as well) and then let then complete the problem. The fifth
example requires using a placeholder for a “missing” term in the dividend.
Example 5: Divide by long division.
Dividend
Divisor
4
3
 10 x  50 x  800
x6
Now is a good time to have the students explore The Remainder Theorem and Factor
Theorem. A good way to do this is to have the students evaluate the dividend in
example 4 ( 2 x 3  5x 2  7 x  20 ) at x  4 . Ask the students what kind of number –4
is. (expect answers like root or zero) Point out that the reminder for the division problem
and the value when x  4 were the same. You can then introduce the Factor Theorem
Factor Theorem
A polynomial f (x) has a factor ( x  k ) if and only if f (k )  0 .
You can now explore if the value of the polynomial at a point, c, is the same as the
remainder when the polynomial is divided by xc is also true if the divisor is not a factor.
Ask the students to evaluate the dividend in example 5 ( 10 x 4  50 x 3  800 ) at x  6 .
Again point out that the reminder for the division problem and the value when x  6
were the same. You can now state the Remainder Theorem
The Remainder Theorem
If a polynomial f (x) is divided by ( x  k ) , the remainder is r  f (k ) .
Polynomials: (45- 60 minutes)
Review:
Ask the students to find the real zeros of the polynomial function f ( x)  x  x  6 .
They should be able to factor this quadratic. Next ask the students to find the real zeros
2
of the polynomial function f ( x)  2 x  11x  15 by factoring. You can also review
factoring a third degree polynomial by finding the real zeros of
2
f ( x)  2 x 3  2 x 2  16 x  16 . In all cases insist that the answers be left in fractional
form.
Material:
Now ask the students if they know how to find the real zeros of the polynomial function
f ( x)  x 3  6 x 2  11x  6 . We need to try something different since we cannot factor
this polynomial. Now draw their attention to the real zeros of the polynomial functions
that were used in the review. Ask them what they notice about the numerators of each of
the zeros in relation to the original polynomials. (Guide them toward the numerators are
factors of the constant term of the polynomial). Also ask them what they notice about the
denominators of the zeros in relation to the original polynomials. Make sure to point out
that integer zeros have a denominator of 1. (Guide them toward the denominators are
factors of the leading coefficient of the original polynomial.) You can now make the
generalization to the Rational Zero Test.
The Rational Zero Test
If the polynomial f ( x )  an x n  an 1 x n 1    a2 x 2  a1 x  a0 has integer
p
coefficients, then every rational zero of f has the form rational zero  where p
q
and q have no common factors other than 1 and p is a factor of the constant term
a0 and q is a factor of the leading coefficient a n .
Now go back to the problem of finding the real zeros of the polynomial function
f ( x)  x 3  6 x 2  11x  6 . Ask the students how we might start to answer this problem
using the new information that we have. (Hopefully they will suggest writing all the
possible rational zeros.) Next ask the students how we might check to see if the possible
rational zeros are really rational zeros. (Most students will tell you to just plug in the
number to check, but you want to encourage them to find out more that just if it is a zero,
but to also factor out the linear factor associated with the zero.) Complete this example
with the students by using polynomial division to factor out each of the three factors
associated with the three rational roots.
Now ask the students to complete example 1 by finding all the possible rational roots and
checking to see if they are really roots.
Example 1: List all the possible rational roots of the polynomial function
f ( x)  2 x 3  3x 2  23 x  12 . Determine which of the possible rational zeros
is actually a zero of the function.
Ask the students to complete example 2.
Example 2: List all the possible rational roots of the polynomial function
f ( x)  2 x 4  x 3  20 x 2  7 x  42 . Determine which of the possible rational
zeros is actually a zero of the function.
This particular example has two rational roots x 
3
and x  2 .
2
Now you can expand example 2 to find all the real zeros. You can point out to the
students that after dividing out the rational zeros, you are left with the quadratic
polynomial expression x  7 . This is a polynomial that the students can find the zeros
of. Ask the students to find all the zeros of the polynomial in example 2.
2
Ask the students what they might do with example 3 (They will likely groan).
Example 3: Find all the real zeros of the polynomial function
f ( x)  10 x 4  77 x 3  8 x 2  159 x  90 .
This example leads well into bypassing the rational zero test and using the
calculator to determine which possible rational zeros to test. Note that this is not
using the calculator’s ability to determine roots, but just looking at the graph to
see what to test and the divide out zeros that work.
While looking at the graph on the calculator, you can also introduce the Intermediate
Value Theorem
Intermediate Value Theorem
Let a and b be real numbers such that a<b. If f is a polynomial function such that
f ( a )  f (b) , then, in the interval [a, b], f takes on every value between f (a ) and
f (b) .
This allows you to say where a zero occurs when f (a ) and f (b) have opposite signs.
This is particularly helpful to those students who do not have a calculator that determines
zeros. You can show all the students how to reset the window and trace to get a good
estimate of the real zeros. Make sure to point out to the students that this theorem is not
just for zeros, but for all values that the function takes on.
The following is a good problem to have the students work in groups.
Example 4: An open box is to be made from a rectangular piece of material, 10
inches by 5 inches, by cutting equal squares from the corners and turning up the sides.
a.
b.
c.
d.
Write the volume V of the box as a function of x.
Find values of x such that V = 56. Which of these values is a physical
impossibility in the construction of the box? Explain.
Determine the domain of the function.
Sketch the graph of the function and approximate the measurements of the box
that yield a maximum volume.
Wrap this up by having the students present their solutions. It sometimes helps to
have the students actually try to “make” the box with a piece of paper to understand
the question.
Complex Numbers: (40 minutes)
Review:
Ask the students to find the real zeros of the polynomial function
f ( x)  x 4  x 3  2 x  4 . They should be able to find 2 real zeros using either the
rational root test and division or by dividing out roots that appear on the graphs of the
calculator and then use the quadratic formula to determine the zeros of the quadratic that
remains.
Material:
The review example leads well into what to do with a negative number under the square
root symbol. It works well to introduce the imaginary unit by looking to the solutions to
x 2  1  0 and x 2  1  0 . In order to answer the second equation, we must introduce
the imaginary unit i   1 .
Now is a good time to go through the powers of i using the definition of i . This allows
the students to work with i in a fairly simple way. After you go through the first 8
powers of i , ask the students if they see a pattern (hopefully someone will come up with
something related to the remainder when the exponent is divided by 4). Next ask the
students to apply their pattern to the following problems. (This can be done in pairs.)
Example 1: Express each of the powers of i as i, -i, 1, or -1
 i 33
 i 20
 i103
 i 74
Now you should introduce the definition of a complex number (in standard form) a  bi .
Ask the students complete following example.
Example 2: Write the complex number in standard form
 1  8

 6i  i 2
Now ask the students how they might add two complex numbers ( (13  2i )  ( 5  6i ) ).
It works will to liken this to adding polynomials where like terms get added. You can do
the same thing with subtracting two complex numbers ( (4  8i)  (8  11i) ) and
multiplying two complex numbers ( (1  i )( 3  2i ) ).
Ask the students to complete the following problems:
Example 3: Perform the indicated operation and write the result in standard
form.
 (3  5i)  (3  3i)




(6  5i)  (4  7i)
(3  5i)(4  3i)
( 2  3i ) 2
(6  7i)(6  7i)
Ask the students to share their answers. You can point out that when adding,
subtractions, or multiplying two complex numbers, it is possible to end up with a number
that is purely imaginary or real. The last problem in the example leads well into the
introduction of the complex conjugate. Make sure to point out that only the sign before
the imaginary part is changed in the complex conjugate.
Ask the students to complete the following example:
Example 4: Write the complex conjugate of each answer from example 3.
Now you need to introduce division of complex numbers. It works well to point out that
division of complex numbers is similar to rationalizing the denominators of fractions in
that the denominator need to be made into a real number.
Work through the example,
5
, with the students. Most students can help you with
4  2i
most of the steps.
Ask the students to complete the following example.
Example 5: Perform the indicated operation and write the result in standard
form.


3  7i
6  2i
(21  7i )(4  3i )
2  5i
Now go back to the question from the beginning but change it slightly to find all the
zeros of the polynomial function f ( x)  x  x  2 x  4 . The students should now
be able to express the answers they found with the quadratic formula as complex numbers
in standard form.
4
3
Ask the students to complete the following example.
Example 6: Solve the equation.


4 x 2  16 x  17  0
9 x 2  6 x  37  0
These topics should be followed up by finding all zeros of polynomials and introducing
the Fundamental Theorem of Algebra and the Linear Factorization Theorem.