The following table lists an nth order polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$ and the forward difference evaluated at equally spaced values of x. The order of the polynomial is

x	f(x)	Δf	Δ²f	Δ³f
-0.4	1.7648	-0.2965	0.089	-0.03
-0.3	1.4683	-0.2075	0.059	-0.0228
-0.2	1.2608	-0.1485	0.0362	-0.0156
-0.1	1.1123	-0.1123	0.0206	-0.0084
0	1	-0.0917	0.0122	-0.0012
0.1	0.9083	-0.0795	0.011	0.006
0.2	0.8288	-0.0685	0.017	0.0132

To determine the order of a polynomial using a difference table, we look for the first set of finite differences that are constant. The order of the polynomial is equal to the order of the constant differences.

Polynomial Order Determination Using Finite Differences

We are given a table of values for x and f(x), along with the calculated forward differences Δf, Δ²f, and Δ³f. Let's analyze these differences to find the order of the polynomial f(x).Analyzing the Forward Differences

We examine the provided difference columns:

- Δf: The values in the Δf column (e.g., -0.2965, -0.2075, -0.1485, etc.) are not constant. This indicates the polynomial is not of the first order (linear).
- Δ²f: The values in the Δ²f column (e.g., 0.089, 0.059, 0.0362, etc.) are also not constant. This means the polynomial is not of the second order (quadratic).
- Δ³f: The values in the Δ³f column are -0.03, -0.0228, -0.0156, -0.0084, -0.0012, 0.0060, and 0.0132. These values are not constant either.

Calculating the Fourth Forward Differences (Δ⁴f)

To determine the polynomial order, we need to calculate the next level of differences, which is the fourth forward difference (Δ⁴f), by taking the differences between consecutive terms in the Δ³f column.

The Δ³f values are: {-0.03, -0.0228, -0.0156, -0.0084, -0.0012, 0.0060, 0.0132}.

Let's calculate Δ⁴f:

- First Δ⁴f: Δ³f(-0.3) - Δ³f(-0.4) = -0.0228 - (-0.03) = 0.0072
- Second Δ⁴f: Δ³f(-0.2) - Δ³f(-0.3) = -0.0156 - (-0.0228) = 0.0072
- Third Δ⁴f: Δ³f(-0.1) - Δ³f(-0.2) = -0.0084 - (-0.0156) = 0.0072
- Fourth Δ⁴f: Δ³f(0.0) - Δ³f(-0.1) = -0.0012 - (-0.0084) = 0.0072
- Fifth Δ⁴f: Δ³f(0.1) - Δ³f(0.0) = 0.0060 - (-0.0012) = 0.0072
- Sixth Δ⁴f: Δ³f(0.2) - Δ³f(0.1) = 0.0132 - 0.0060 = 0.0072

The calculated Δ⁴f values are {0.0072, 0.0072, 0.0072, 0.0072, 0.0072, 0.0072}.

Conclusion

Since the fourth forward differences (Δ⁴f) are constant (all equal to 0.0072), the polynomial is of the fourth order.