To find the \( x^2 \) term in the expansion of \( (x - 5)^4 \) using the Binomial Theorem, we can use the formula:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In this case, \( a = x \), \( b = -5 \), and \( n = 4 \).
The general term in the expansion can be written as:
\[ T_k = \binom{n}{k} a^{n-k} b^k = \binom{4}{k} x^{4-k} (-5)^k \]
We want the coefficient of \( x^2 \), which corresponds to \( 4 - k = 2 \) or \( k = 2 \).
Now, we can calculate the term for \( k = 2 \):
\[ T_2 = \binom{4}{2} x^{4-2} (-5)^2 \]
First, we calculate the binomial coefficient:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
Next, we calculate \( (-5)^2 \):
\[ (-5)^2 = 25 \]
Now, substitute these values into the term to find \( T_2 \):
\[ T_2 = 6 x^2 \cdot 25 = 150 x^2 \]
Thus, the \( x^2 \) term in the expansion of \( (x - 5)^4 \) is \( 150x^2 \).
The correct response is:
150 x squared
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