Question 1 (7 points)
Each table of values represents a polynomial function. Usc Anite differences to determine the following for each: i the degree
i the sign of the leading coefficient it the value of the leading coefficient
124
- 2
41
8
+31
- 104
-247
To determine the degree, sign of the leading coefficient, and value of the leading coefficient of each polynomial function, we need to examine the differences of the values in the table. If the second differences are constant, then it is a polynomial of degree 2. If the third differences are constant, then it is a polynomial of degree 3, and so on.
Let's calculate the differences for each table:
Table 1:
124 -2 41 8
-126 43 -33
169 -76
-245
Table 1 has constant fourth differences, which means it is a polynomial of degree 4.
Degree: 4
To determine the sign and value of the leading coefficient, we look at the highest degree term (degree 4 term). Since the differences show a decrease from 124 to -2, the leading coefficient is negative.
Leading coefficient sign: Negative
The leading coefficient is the coefficient of the highest degree term, which in this case is the degree 4 term. The value of the leading coefficient can be obtained from the first term in the differences sequence: 124. This is the value of the leading coefficient.
Leading coefficient value: 124
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Table 2:
+31 -104 -247
-135 -251
-116
Table 2 has constant third differences, which means it is a polynomial of degree 3.
Degree: 3
The sign of the leading coefficient can be determined by the overall trend of the values, which in this case is a decrease. Therefore, the leading coefficient is negative.
Leading coefficient sign: Negative
The leading coefficient is the coefficient of the highest degree term, which in this case is the degree 3 term. The value of the leading coefficient can be obtained from the first term in the differences sequence: 31. This is the value of the leading coefficient.
Leading coefficient value: 31