Question

jaime applied the binomial theorem for the expansion of (x-5)^4 which of the following is the x^2 term
A.150x^2
B.-150x^2

Answer 1

To find the \( x^2 \) term in the expansion of \( (x - 5)^4 \) using the binomial theorem, we start with the binomial expansion formula:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case, we have \( a = x \), \( b = -5 \), and \( n = 4 \):

\[
(x - 5)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} (-5)^k
\]

To get the \( x^2 \) term, we need to find the term where \( 4 - k = 2 \), which means \( k = 2 \).

Now we will calculate that term:

\[
\text{For } k = 2:
\]

\[
\binom{4}{2} x^{4-2} (-5)^2
\]

Calculating each component:

- \( \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \)
- \( (-5)^2 = 25 \)

Now we substitute these values into the term:

\[
6 \cdot x^{2} \cdot 25 = 150x^2
\]

Thus, the \( x^2 \) term in the expansion of \( (x - 5)^4 \) is:

\[
150x^2
\]

The answer is \(\text{A. } 150x^2\).