Use the image to answer the question.

Question

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.

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

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

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

An3+Bn2+Cn+D

Answer 1

To determine the standard form of the polynomial expression based on the successive differences of the sequence, we need to analyze the number of rows (or layers of differences) shown in the image.

In general:

• If the first differences (1st row) are constant, the polynomial is of degree 1 (linear).
• If the second differences (2nd row) are constant, the polynomial is of degree 2 (quadratic).
• If the third differences (3rd row) are constant, the polynomial is of degree 3 (cubic).
• If the fourth differences (4th row) are constant, the polynomial is of degree 4 (quartic).

If the image indicates a constant difference in the 4th row, it suggests that the polynomial is of degree 4.

Based on this reasoning, the correct expression would be:

An^4 + Bn^3 + Cn^2 + Dn + E

So the final answer is:

An^4 + Bn^3 + Cn^2 + Dn + E