There are four rows of numbers with horizontal lines between the numbers and vertical lines extending to the numbers in the row below, indicating relationships.

Question

The successive differences of a sequence are shown in the image. What is the standard form of its related polynomial expression?

(1 point)
Responses

An2+Bn+C
upper A n squared plus upper B n plus upper C

An4+Bn3+Cn2+Dn+E
upper A n superscript 4 baseline plus upper B n cubed plus upper C n squared plus upper D n plus E

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
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Answer 1

To determine the standard form of the related polynomial expression based on the succession of differences provided, we need to identify the degree of the polynomial.

1. If the first differences of the values are constant, the sequence can be modeled by a linear polynomial of the form \( An + B \).
2. If the second differences are constant, the sequence corresponds to a quadratic polynomial \( An^2 + Bn + C \).
3. If the third differences are constant, it corresponds to a cubic polynomial \( An^3 + Bn^2 + Cn + D \).
4. If the fourth differences are constant, it corresponds to a quartic polynomial \( An^4 + Bn^3 + Cn^2 + Dn + E \).

Since the question refers to "successive differences," we would look at how many levels of differences need to be calculated before they become constant. Without the actual numerical data to analyze, I can't provide a definitive answer.

However, the correct standard form is based on the degree of the polynomial that corresponds to the level at which the differences become constant. You would choose an answer based on your findings from those differences:

• If first differences are constant: \( An + B \)
• If second differences are constant: \( An^2 + Bn + C \)
• If third differences are constant: \( An^3 + Bn^2 + Cn + D \)
• If fourth differences are constant: \( An^4 + Bn^3 + Cn^2 + Dn + E \)

Once you identify the degree using the differences, select the corresponding polynomial expression.