A diagram shows four rows with numbers and blanks. Horizontal lines are shown between each number or blank in a row, and vertical lines extend from the horizontal lines to the numbers or blanks in the row below, indicating a relationship.

Question

The successive differences of a sequence are shown in this image. What is the degree of its associated polynomial expression?

Answer 1

To determine the degree of the polynomial associated with a sequence of numbers based on successive differences, we can follow these steps:

1. Compute the successive differences: The first step involves calculating the differences between consecutive terms of the sequence. The result is known as the "first difference."

2. Continue computing differences: If the first differences are not constant, compute the second differences (differences of the first differences). If the second differences are also not constant, compute the third differences, and so on.

3. Check for constancy: The degree of the polynomial is determined by the first level of differences that becomes constant.

• If the first differences are constant, then the original sequence can be represented by a polynomial of degree 1 (a linear polynomial).
• If the second differences are constant, it indicates a degree 2 polynomial (a quadratic polynomial).
• If the third differences are constant, it indicates a degree 3 polynomial (a cubic polynomial).
• This pattern continues, where the n-th level of differences being constant indicates a polynomial of degree n.

By following these steps, you can determine the degree of the polynomial associated with the given sequence of numbers. Examine the provided diagram and calculating differences accordingly to establish which level becomes constant.

If you have specific numbers from the diagram, you can share them to illustrate the process more accurately.