Let's
explore the roots of the following equation:
ax^3
+ bx^2 + cx + d = 0
Before
we begin looking at graphs, consider the following questions:
 How many
roots will this equation have?
 What
kinds of roots (real, imaginary, complex) can this equation have?
Now, we will begin
exploring how the values of each of the coefficients a, b, c, and d relate
to the roots of the equation. We can do this by looking at the graphs
in each of the following planes:
 The xb plane
 The xc plane
 The xd plane
 The xa plane
How can
we explore the equations in these planes using graphing calculator?
(Just curious to see if the class can come up with the idea of replacing
the coefficient with y before I introduce the idea.)
Below is a graph
of ax^3 + bx^2 + y x + d = 0; (a
= 1, b=1, and c=1)
The
red line represents y=2. Describe the roots of the equation when
y = 2. How many roots are represented by this graph when y=2? Are there
roots of the equation not represented by this graph when y=2?
The
green line represents y = 3. Revisit the above questions when
y=3
How
would you interpret the blue line on the graph?
Below is a graph
of ax^3 + yx^2 + cx + d = 0; (a
= 1, c=1, and d=1)
The
red line represents y=2. Describe the roots of the equation when
y = 2. How many roots are represented by this graph when y=2? Are there
roots of the equation not represented by this graph when y=2?
The
green line represents y = 3. Revisit the above questions when
y=3
How
would you interpret the blue line on the graph?
Below is a graph
of ax^3 + bx^2 + cx + y = 0; (a
= 1, b=1, and c=1)
In
this graph no matter what the value of y, y =d only crosses the graph
onceHow do you interpret this fact?
What
should we do to explore this situation more?
(Hopefully, a student will suggest changing the values of a, b, and c.)
Below is a graph
of ax^3 + yx^2 + cx + d = 0; (a
= 1, c=2, and d=3)
For
what values of d does the equation x^3 +2x^2 + 3x + d = 0 have three
real roots?
Can
this equationor any third degree equation with integer coefficientsever
have exactly two real roots? Explain why or why not.
(This
question should lead to the idea of one of the roots being a "double"
rootand that is the only case where this can happen. In particular,
I want them to see that one complex root and two real roots won't lead
to an equation with integer coefficients)
Below
is a graph of
yx^3 + bx^2 + cx + d = 0;
It
appears that when y = .5, that the equation has exactly two distinct real
rootsis this possible?
How
can we explain the way this graph appears? How many times do you think
the line y = .5 actually intersects this graph?
