To show:The third differences of a polynomial function of degree 3 are nonzero and constant.
Given information:
Given function is
Calculation:
Substitute different values of
Substitute
Now Substitute
Similarly, more values can be substituted, and the corresponding value will be found. The table for different values is given below:
3 | -12 |
2 | -14 |
1 | -10 |
0 | -6 |
-1 | -8 |
-2 | -22 |
-3 | -54 |
Now first difference can be found by subtracting 2 digits from first digit and so on.
This gives:
First difference | Second difference | Third difference | ||
3 | -12 | 4-2=2 | 0-2=-2 | |
2 | -14 | 4-4=0 | ||
1 | -10 | -2-4=-6 | ||
0 | -6 | |||
-1 | -8 | |||
-2 | -22 | |||
-3 | -54 |
It is clear from the table that third difference is nonzero and constant.
Now take the polynomial
Substitute different values of
Substitute
Now Substitute
Similarly, more values can be substituted, and the corresponding value will be found. The table for different values is given below:
3 | |
2 | |
1 | |
0 | |
-1 | |
-2 | |
-3 |
Now first difference can be found by subtracting 2 digits from first digit and so on. This gives:
First difference | Second difference | Third difference | ||
3 | ||||
2 | ||||
1 | ||||
0 | ||||
-1 | ||||
-2 | ||||
-3 |
It is clear from the table that third difference is nonzero and constant.
Chapter 5 Solutions
High School Math 2015 Common Core Algebra 2 Student Edition Grades 10/11
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