Assignment:Correcting Polynomial Root Responses

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Assignment: Correcting Polynomial Root Responses
Part 1 – Who’s Right?
Each of the following questions has been answered by two students, Edgar and Emily. In each
case, one student’s answer is correct, and the other student’s answer is incorrect. Decide which
student is right and then describe the other student’s misconceptions.
For each question:
a. State which student’s response is correct.
b. Identify and correct the misconceptions in the incorrect student’s response.
c. Write 2-3 sentences describing the incorrect student’s mistakes and how it should be
corrected.
Problem 1 is an example. Use it as a guide for answering the problems that follow.
1. Write a polynomial equation of the smallest degree with roots 1, 5, and -2.
Edgar Emily r1 1, r2 5, r3 2 r1 1, r2 5, r3 2 x 1 x 5 x 2 0 x 1 x 2 2x 5 x 10 0 x 1 x 2 3 x 10 0 x 1 x 5 x 2 0 x 1 x 2 2x 5 x 10 0 x 1 x 2 3 x 10 0 x 3 3 x 2 10 x x 2 3 x 10 0 x 3 3 x 2 10 x x 2 3 x 10 0 x 3 4 x 2 7 x 10 0 x 3 2 x 2 13 x 10 0 f ( x ) x 3 4 x 2 7 x 10 f ( x ) x 3 2 x 2 13 x 10 Answers:
a. Emily’s answer is incorrect.
b. Emily made the following mistake: x 1 x 5 x 2 0
The corrected answer is: x 1 x 5 x 2 0
c. Emily’s mistake was in changing the roots of the polynomial into factors. When writing the
factors, it is important to take the opposite of the root. For example, the first root was r1 1 .
This makes the corresponding root x 1 . © K12 Inc. 2. Write a polynomial equation of the smallest degree with roots 3, 4i, and -4i.
Edgar Emily r1 3, r2 4i , r3 4i r1 3, r2 4i , r3 4i x 3 x 4i x 4i 0 x 3 x 2 4ix 4ix 16i 2 0 x 3 x 2 16 0 x 3 x 4i x 4i 0 x 3 x 2 4ix 4ix 16i 2 0 x 3 x 2 16 0 x 3 16 x 3 x 2 48 0 x 3 16 x 3 x 2 48 0 f ( x ) x 3 3 x 2 16 x 48 f ( x ) x 3 3 x 2 16 x 48 3. Identify the roots for the equation x 2 x 6i x 6i 0 .
Edgar Emily x 2 x 6i x 6i 0 x 2 x 6i x 6i 0 x 2 0 x20 x2 x2 x 6i 0 x 6i 0 x 6i x 6i x 6i 0 x 6i 0 x 6i x 6i r1 2, r2 6i , r3 6i r1 2, r2 6i , r3 6i © K12 Inc. 4. Identify the zeros of the function graphed below. Edgar Emily The zeros are x = -2, x = 0, and x = 1. The zeros are x = -1.25 and x = 0.5. © K12 Inc. 5. Write a polynomial equation of the smallest degree with roots 0, 5, and -2.
Edgar Emily r1 0, r2 5, r3 2 r1 0, r2 5, r3 2 x 0 x 5 x 2 0 x x x x 2 2 x 5 x 10 0
2 3 x 10 0 x 3 3 x 2 10 x 0 x 0 x 5 x 2 0 x 2 2 x 5 x 10 0 x 2 3 x 10 0
f ( x ) x 2 3 x 10 f ( x ) x 3 3 x 2 10 x © K12 Inc. Part 2 – Common Mistakes
The problems above describe some of the most common mistakes made when first studying the
roots of polynomial equations and using the fundamental theorem of algebra.
In your opinion, which mistake do you think occurs most often? Which mistake will you be most
cautious of as you complete these types of problems? In your response be sure to include the
following: Explain and tell why you think a particular mistake occurs the most.
Explain and tell why you think a particular mistake is the easiest to make.
What steps can you take that will help you avoid these types of mistakes in the future? © K12 Inc.

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