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The successive differences of a sequence are shown in the image. What is the standard form of its related polynomial expression? (1 point) Responses An+B upper A n plus upper B An3+Bn2+Cn+D upper A n cubed plus upper B n squared plus upper C n plus upper D An2+Bn+C upper A n squared plus upper B n plus upper C An4+Bn3+Cn2+Dn+E

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To determine the polynomial expression related to the successive differences of a sequence, we generally follow these steps:

  1. Identify the degree: The number of times you need to take differences until you reach a constant indicates the degree of the polynomial. If the k-th differences are constant, then the original sequence is generated by a polynomial of degree k.

  2. Construct the polynomial expression: Based on the degree identified, we can choose the corresponding polynomial form.

Since I'm unable to see the image, I can't directly assess the number of differences and their nature. However, I can help you decide among the options provided based on the common relationship between differences and polynomials:

  • If the first differences are constant, it indicates a linear polynomial of the form \( An + B \).
  • If the second differences are constant, it indicates a quadratic polynomial of the form \( An^2 + Bn + C \).
  • If the third differences are constant, it indicates a cubic polynomial of the form \( An^3 + Bn^2 + Cn + D \).
  • If the fourth differences are constant, it indicates a quartic polynomial of the form \( An^4 + Bn^3 + Cn^2 + Dn + E \).

Please analyze the differences in the image to identify the level of constant differences, which will help you choose the correct polynomial form from the list provided. If you can share details or the number of differences you noted, I would be able to guide you more accurately.