To determine who is correct, let's analyze the claims made by Bruno and Juliana.
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Bruno's Claim: He states that the sequence has 5th constant differences of 240. This implies that the sequence can indeed be represented by a polynomial of degree 5, as a sequence with constant differences of order \( n \) corresponds to a polynomial of degree \( n \). Therefore, Bruno is correct in saying that the expression is a 5th-degree polynomial.
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Juliana's Claim: She suggests that the leading term of the polynomial has a coefficient of 48. The leading term of a 5th-degree polynomial can be derived from the 5th difference constant. The relationship between the leading coefficient \( a_5 \) of a polynomial of degree 5 and the 5th difference \( D_5 \) is given by:
\[ D_5 = 5! \cdot a_5 \]
Since \( D_5 = 240 \), we can solve for \( a_5 \):
\[ 240 = 120 \cdot a_5 \quad \Rightarrow \quad a_5 = \frac{240}{120} = 2 \]
Therefore, the leading coefficient \( a_5 \) is 2, not 48. Hence, Juliana's claim about the leading term with a coefficient of 48 is incorrect.
Based on this analysis:
- Bruno is correct in stating that the expression is a 5th-degree polynomial.
- Juliana is incorrect about the leading term's coefficient.
Thus, the correct response is: only Bruno.
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