SITE-A teams: Lesson 20

 

A cubic polynomial presents a distinctive graphic signature when its graph is drawn. The size (fat or thin), direction (opening up or opening down), and position on the xy-plane vary but the curve is a pattern that students can recognize. Reference: Lopez, A. (1996) Pattern Matching, Searching and Heuristics in Algebra, Mathematics and Computer Education, 30, 2, 255-266.

Algebra students are often faced with three types of problems dealing with cubic polynomials:

    1. Find the real-valued roots (x-intercepts) of a given cubic polynomial.
    2. Given a cubic polynomial equal to zero, solve for the real-values of x.
    3. Factor a given cubic polynomial.

All three of these activities are closely related and can be accomplished graphically using the CASIO fx7400G.

Consider the following:
    1. Find the roots of         y = x3 - 4 x2 + 7 x - 6
    2. Solve for x:                 x3 - 4 x2 + 7 x - 6 = 0
    3. Factor                         x3 - 4 x2 + 7 x - 6

Turn on the CASIO fx7400G.

Ac_on.gif (1767 bytes)

Use your cursor arrows to highlight the GRAPH menu and press exe.gif (1747 bytes) or just press 4.gif (1703 bytes) when at the main menu screen.

In the Y1 slot enter the expression X3 - 4 X2 + 7 X - 6 and press exe.gif (1747 bytes)

(Make sure no other Y-slot is highlighted, i.e. selected for graphing.)

Press shiftf3.gif (1888 bytes) (V-Window)

Press f1.gif (1719 bytes) (INIT)

Press quit.gif (1764 bytes)

Press f4.gif (1732 bytes) (DRAW)

The graph appears on the screen of the calculator (See graph below). Press shiftf1.gif (1869 bytes) (Trace) and use the right cursor arrow to move the "cross hairs" to where the graph touches the x-axis. This is at x = 2 and y = 0.

Since the graph crosses the x-axis once, the answers are:

    1. there is one real-valued root (x-intercept) and it is 2.
    2. the real-valued solution is x = 2.
    3. the polynomial is factored into ( x - 2 )( x2 - 2 x + 3 )

Press quit.gif (1764 bytes)

Highlight the expression and de-select it by pressing f1.gif (1719 bytes) (SEL)

or delete it by pressing f2.gif (1726 bytes) (DEL)

Consider the following:
    1. Find the roots of        y = x3 - x2 - x + 1
    2. Solve for x:                  x3 - x2 - x + 1 = 0
    3. Factor                         x3 - x2 - x + 1

In the Y2 slot enter the expression X3 - X2 - X + 1 and press exe.gif (1747 bytes)

(Make sure no other Y-slot is highlighted, i.e. selected for graphing.)

Press f4.gif (1732 bytes) (DRAW)

The graph appears on the screen of the calculator (See graph below). Press shiftf1.gif (1869 bytes) (Trace) and use the right cursor arrow to move the "cross hairs" to where the graph touches the x-axis. This is at x = - 1 and y = 0, and at x = 1 and y = 0.

Since the graph crosses the x-axis in one places and just touches it in the other, the answers are:

    1. the real-valued roots (x-intercepts) are - 1 and 1.
    2. the real-valued solutions to the equation are x = - 1 and x = 1
    3. the polynomial factors into ( x + 1)( x - 1 )2.

Press quit.gif (1764 bytes)

Highlight the expression and de-select it by pressing f1.gif (1719 bytes) (SEL) or delete it by pressing f2.gif (1726 bytes) (DEL)

Consider the following:
    1. Find the roots of        y = - x3 - x2 + 4 x + 4
    2. Solve for x:                 - x3 - x2 + 4 x + 4 = 0
    3. Factor                        - x3 - x2 + 4 x + 4

In the Y3 slot enter the expression - X3 - X2 + 4 X + 4 and press exe.gif (1747 bytes)

(Make sure no other Y-slot is highlighted, i.e. selected for graphing.)

Press f4.gif (1732 bytes) (DRAW)

The graph appears on the screen of the calculator (See graph below). Press shiftf1.gif (1869 bytes) (Trace) and use the right cursor arrow to move the "cross hairs" to where the graph touches the x-axis. This is at x = -2 and y = 0, at x = -1 and y = 0, and at x = 2 and y = 0.

Since the graph crosses the x-axis in three places, the answers are:

    1. the real-valued roots (x-intercepts) are -2, -1 and 2.
    2. the solutions to the equation are x = -2, x = -1, and x = 2.
    3. the polynomial factors into ( x + 2 )( x + 1 )( x - 2 ).

Press quit.gif (1764 bytes)

Highlight the expression and de-select it by pressing f1.gif (1719 bytes) (SEL) or delete it by pressing f2.gif (1726 bytes) (DEL)

Press menu.gif (1772 bytes)

Turn off the CASIO fx7400G. shiftac.gif (1924 bytes) (OFF)

 

lesson20a.gif (2898 bytes)

Graph of y = x3 - 4 x2 + 7 x - 6

lesson20b.gif (3134 bytes)

Graph of y = x3 - x2 - x + 1

lesson20c.gif (3270 bytes)

Graph of y = - x3 - x2 + 4 x + 4