- Calculate the first-order differences of the given \(f(x)\) values to identify the change rate.
- Compute subsequent orders of differences (second, third, and fourth) by subtracting consecutive terms of the previous difference list.
- Observe that the fourth-order differences are all equal to 24, confirming that the function is a fourth-degree polynomial.
- Conclude that the degree of the polynomial \(f(x)\) is 4. The final answer is \(\boxed{\text{Fourth Degree}}\).
Explanation
- Analysis of Data and Method Selection
To determine the degree of the polynomial function \(f(x)\), we examine the given table of values and utilize the method of finite differences. This method states that for a polynomial of degree \(n\), the \(n\)-th order differences are constant, while higher-order differences are zero.
The provided data points \((x, f(x))\) are:
- \(f(-4) = 366\)
- \(f(-3) = 141\)
- \(f(-2) = 36\)
- \(f(-1) = -9\)
- \(f(0) = -30\)
- \(f(1) = -39\)
- \(f(2) = -24\)
- \(f(3) = 51\)
- \(f(4) = 246\)
- Calculating First-Order Differences
We calculate the differences between consecutive \(f(x)\) values, which is the first-order difference \(\Delta_1\):
- \(141 - 366 = -225\)
- \(36 - 141 = -105\)
- \(-9 - 36 = -45\)
- \(-30 - (-9) = -21\)
- \(-39 - (-30) = -9\)
- \(-24 - (-39) = 15\)
- \(51 - (-24) = 75\)
- \(246 - 51 = 195\)
Thus, \(\Delta_1 = [-225, -105, -45, -21, -9, 15, 75, 195]\).
- Calculating Second-Order Differences
Next, we calculate the second-order differences \(\Delta_2\) by finding the differences of consecutive values in \(\Delta_1\):
- \(-105 - (-225) = 120\)
- \(-45 - (-105) = 60\)
- \(-21 - (-45) = 24\)
- \(-9 - (-21) = 12\)
- \(15 - (-9) = 24\)
- \(75 - 15 = 60\)
- \(195 - 75 = 120\)
Thus, \(\Delta_2 = [120, 60, 24, 12, 24, 60, 120]\).
- Calculating Third-Order Differences
We calculate the third-order differences \(\Delta_3\) by finding the differences of consecutive values in \(\Delta_2\):
- \(60 - 120 = -60\)
- \(24 - 60 = -36\)
- \(12 - 24 = -12\)
- \(24 - 12 = 12\)
- \(60 - 24 = 36\)
- \(120 - 60 = 60\)
Thus, \(\Delta_3 = [-60, -36, -12, 12, 36, 60]\).
- Calculating Fourth-Order Differences and Conclusion
Finally, we calculate the fourth-order differences \(\Delta_4\) by finding the differences of consecutive values in \(\Delta_3\):
- \(-36 - (-60) = 24\)
- \(-12 - (-36) = 24\)
- \(12 - (-12) = 24\)
- \(36 - 12 = 24\)
- \(60 - 36 = 24\)
The fourth-order differences are constant at 24. Since the fourth-order differences are constant, the polynomial \(f(x)\) is a fourth-degree polynomial. Thus, the correct degree is 4. The final answer is \(\boxed{\text{Fourth Degree}}\).
Examples
The method of finite differences is widely used in data analysis and computer science to determine the underlying relationship in data. For instance, if you are tracking the speed of a vehicle and you have data points for its position over time at constant intervals, finite differences can help you determine if the vehicle is moving at a constant velocity (first degree), constant acceleration (second degree), or changing acceleration (third degree or higher). This is a fundamental concept in modeling physical systems and predicting future outcomes based on historical data.