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Answer:
3
Step-by-step explanation:
The x-values in the table are evenly-spaced, so we can determine the degree of the function by looking at successive differences of the f(x) terms.
f(x): -23, -4, 3, 4, 5, 12, 31
first differences: 19, 7, 1, 1, 7, 19
second differences: -12, -6, 0, 6, 12
third differences: 6, 6, 6, 6
The third differences are constant, so the function values can be matched by a third degree polynomial.
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Additional comment
The first differences are found by subtracting the function value from the next one. Likewise, the second differences are found by subtracting each first-difference value from the next one. An so on for the third differences.
The level at which the differences are constant is the degree of the polynomial that will match the function values. (Compare this to what you know about an arithmetic sequence, where first differences are constant and the sequence is described by a linear (degree 1) equation.)