Writing a Polynomial Function from its Given Zeros, Lecture notes of Elementary Mathematics

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Writing a

Polynomial Function

from its Given Zeros

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Goals and Objectives

• Students will be able to write a polynomial from its given zeros.

117 Write the polynomial function of lowest degree using the

zeros given.

A

B

C

D

x = -.5, multiplicity of 1 x = 3, multiplicity of 1 x = 2.5, multiplicity of 1

Answer

118 Write the polynomial function of lowest degree using the

zeros given.

A

B

C

D

x = 1/3, multiplicity of 1 x = -2, multiplicity of 1 x = 2, multiplicity of 1

Answer

Write the polynomial function of lowest degree using the zeros from the given graph, including any multiplicities.

x = -

x = -

x = 1.5 (^) x = 3

x = -

x = -

x = 1.

x = 3

or (^) or or

120 Write the polynomial function of lowest degree using the

zeros from the given graph, including any multiplicities.

A B C D E F

Answer

122 Which equation could be the equation of the graph

below?

A

B

C

D

Answer

Match each graph to its equation.

y = x^2 + 2

y = ( x + 2)^2

y = ( x -1)( x -2)( x -3)

y = ( x -1)( x -2)( x -3) 2 Answer

Analyzing Graphs using a Graphing Calculator

Enter the function into the calculator (Hit y= then type).

Check your graph, then set the window so that you can see the zeros and the relative minima and maxima. (Look at the table to see what the min and max values of x and y should be.)

Use the Calc functions ( 2nd TRACE ) to find zeros:

Select 2: Zero Your graph should appear. The question "Left Bound?" should be at the bottom of the screen.

Use the left arrow to move the blinking cursor to the left side of the zero and press ENTER. The question "Right Bound?" should be at the bottom of the screen.

Use the right arrow to move the blinking cursor to the right side of the zero and press ENTER. The question "Guess?" should be at the bottom of the screen.

Press ENTER again, and the coordinates of the zero will be given.

Use the Calc functions (2nd TRACE) to find relative min or max:

Select 3: minimum or 4: maximum. Your graph should appear. The question "Left Bound?" should be at the bottom of the screen.

Use the left arrow to move the blinking cursor to the left side of the turning point and press ENTER. The question "Right Bound?" should be at the bottom of the screen.

Use the right arrow to move the blinking cursor to the right side of the turning point and press ENTER. The question "Guess?" should be at the bottom of the screen.

Press ENTER again, and the coordinates of the min or max will be given.

Finding Minima and Maxima

Use a graphing calculator to find the zeros and turning points of

Answer

Sketch the graph of f ( x ) = ( x -1)( x +1)( x -2)( x +2)( x -3)( x +3)( x -4). After sketching, click on the graph to see how accurate your sketch is.

An open box is to be made from a square piece of cardboard that measures 50 inches on a side by cutting congruent squares of side- length x from each corner and folding the sides.

50

x

1. Write the equation of a polynomial function to represent the volume of the completed box.
2. Use a graphing calculator or graphing utility to create a table of values for the height of the box. (Consider what the domain of x would be.) Use the table to determine what height will yield the maximum volume.
3. Look at the graph and calculate the maximum volume within the defined domain. Does this answer match your answer above? (Use the table values to x determine how to set the viewing window.)

Answer

1.

An engineer came up with the following equation to represent the height, h ( x ), of a roller coaster during the first 300 yards of the ride: h ( x ) = -3 x^4 + 21 x^3 - 48 x^2 + 36 x, where x represents the horizontal distance of the roller coaster from its starting place, measured in 100's of yards. Using a graphing calculator or a graphing utility, graph the function on the interval 0 < x < 3. Sketch the graph below.

Does this roller coaster look like it would be fun? Why or why not?

Answer

(^ Derived from (