Question

Jaime applies the Binomial Theorem for the expansion of (x−5)4 . Which of the following is the x2 term?(1 point) Responses −30x2 negative 30 x squared 150x2 150 x squared −150x2 negative 150 x squared 6x2

Answer 1

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**