Question

Jaime applies the Binomial Theorem for the expansion of (x−5)^4 . Which of the following is the x^2 term?

Answer 1

To find the \(x^2\) term in the expansion of \((x - 5)^4\) using the Binomial Theorem, we can apply the theorem which states:

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

In this case, \(a = x\), \(b = -5\), and \(n = 4\). We want to find the term that contains \(x^2\), which means we need to find the term where \(n-k=2\) (i.e., the exponent of \(x\) is 2).

Setting \(n-k = 2\):

\[
4 - k = 2 \implies k = 2
\]

Now, we can find the desired term when \(k = 2\):

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

Calculating \( \binom{4}{2} \):

\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \cdot 3}{2 \cdot 1} = 6
\]

Now, compute \( (-5)^2 \):

\[
(-5)^2 = 25
\]

Substituting back, we find the \(x^2\) term:

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

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

\[
\boxed{150x^2}
\]