Question

Geetha applies the Binomial Theorem for the expansion of (3x−5y)5
. Which of the following is the last term in the expression?(1 point)
Responses

−5y5
negative 5 y superscript 5 baseline

−3,125y5
negative 3,125 y superscript 5 baseline

3,125y5
3,125 y superscript 5 baseline

5y5

Answer 1

To find the last term in the expansion of \((3x - 5y)^5\) using the Binomial Theorem, we can consider the general term in the binomial expansion, which is given by:

\[
T_k = \binom{n}{k} a^{n-k} b^k
\]

where \( n \) is the power, \( a \) and \( b \) are the terms in the binomial, and \( k \) is the index of the term.

Here, \( a = 3x \), \( b = -5y \), and \( n = 5 \).

The last term in the expansion corresponds to \( k = n \), so we will calculate the term \( T_5 \):

\[
T_5 = \binom{5}{5} (3x)^{5-5} (-5y)^5
\]

Calculating each component:

- \(\binom{5}{5} = 1\)
- \((3x)^{0} = 1\)
- \((-5y)^5 = -5^5 y^5 = -3125 y^5\)

Putting it all together:

\[
T_5 = 1 \cdot 1 \cdot (-3125y^5) = -3125 y^5
\]

Thus, the last term in the expression is:

\(-3125 y^5\)

So the correct response is:

**−3,125y5**