Question

Geetha applies the Binomial Theorem for the expansion (3x−5y)^5 . Which of the following is the last term in the expression.(1 point)
-5y^5
5y^5
3,125y^5
-3,125y^5

Answer 1

In the Binomial Theorem, the general term in the expansion of \((a + b)^n\) is given by:

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

In this case, we have \(a = 3x\), \(b = -5y\), and \(n = 5\). The last term in the expansion corresponds to \(k = n\).

For the given expression \((3x - 5y)^5\), the last term is when \(k = 5\):

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

Calculating this:

\[
T_5 = \binom{5}{5} (3x)^0 (-5y)^5
\]

\(\binom{5}{5} = 1\) and \((3x)^0 = 1\), thus:

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

Now, we calculate \((-5y)^5\):

\[
(-5y)^5 = -5^5 y^5 = -3125y^5
\]

Therefore, the last term in the expansion of \((3x - 5y)^5\) is:

\[
\boxed{-3125y^5}
\]