# A contradiction can be defined as a statement that is necessarily false

August 31, 2017

Question
A contradiction can be defined as a statement that is necessarily false. As you study mathematics, especially logic, you will examine contradictory statements.

Follow the directions outlined in the project guide. For this assignment you will be given three statements about polynomials. Two of the statements will agree with each other, while one of the statements will contradict the other two. It will be up to you to identify the contradictory statement and explain how it can be changed so that it is agrees with the other two statements.

Assignment: Roots of Polynomial Functions
Part 1 – Identifying the Contradictory Statement
In each of the following problems, three statements are given. Two of the statements agree with
For each problem:
Explain why the statement you chose contradicts each of the other two.
Describe how the contradictory statement could be changed to make it agree with the
other two statements.

1.
Statement 1:
Statement 2:

4 is a root of the polynomial f ( x )
3

Statement 3:

2.
Statement 1:

Statement 2:

Statement 3:

f ( x ) x 7x
f ( 4)
30

x3

7x

6.

6

The polynomial f ( x )

2x 3

x2

25 x 12 has exactly three rational roots:

1
x
3, x
,x 4
2
.
According to the rational root theorem, the possible rational roots of the
1
3
, 1
,
, 2, 3, 4, 6,
polynomial f ( x ) 2x 3 x 2 25 x 12 are
2
2
5 2 1 25 12

10

By synthetic division:

2

45

12

9

7

0

12 .

3.
Statement 1:

The graph of the function f ( x )

Statement 2:

f ( x ) 2x 3
f (1) 6
f (2)
6

Statement 3:

x

9x 2

2x 3

9x 2

x 12 is:

x 12

3
is a rational root of the function f ( x )
2

2x 3

9x 2

x 12 .

Part 2 – Write Your Own Problem
Now it’s your turn to write a problem. For the problem you will:
Write three statements about roots of polynomials. Two of the statements should agree
with each other. The third statement should contradict the other two.