Final answer:
The binomial expansion of (3x+5)⁹ has 10 terms, as determined by adding 1 to the exponent in the binomial theorem.
option D is correct answer.
Step-by-step explanation:
The question asks how many terms are in the binomial expansion of (3x+5)⁹. The number of terms in the expansion of (a+b)⁹ according to the Binomial Theorem is given by the exponent n plus 1. Therefore, the expansion will have 9+1 terms, which makes the answer d. 10.
The binomial expansion of
(
3
�
+
5
)
9
(3x+5)
9
can be determined using the binomial theorem. The general form of the binomial theorem for
(
�
+
�
)
�
(a+b)
n
states that the expansion will have
�
+
1
n+1 terms. In this specific case, where
(
3
�
+
5
)
9
(3x+5)
9
is expanded, there will be
9
+
1
=
10
9+1=10 terms in the expansion.
Each term in the expansion is obtained by raising the first term,
3
�
3x, to decreasing powers from 9 down to 0, while the second term, 5, is raised to increasing powers from 0 up to 9. The coefficients for each term are determined by binomial coefficients.
So, the binomial expansion of
(
3
�
+
5
)
9
(3x+5)
9
will consist of 10 terms, each with a combination of powers of
3
�
3x and 5, and the coefficients derived from the binomial coefficients.