The binomial theorem provides a way to expand expressions of the form [tex]\((a + b)^n\)[/tex]. According to the binomial theorem, the expanded form is given by:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k,
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\binom{n}{k} = \frac{n!}{k! (n-k)!}\)[/tex].
### Part A: Determine the first term of the binomial expansion [tex]\((x + y)^4\)[/tex].
The first term in the binomial expansion corresponds to [tex]\(k = 0\)[/tex]. So, we take:
[tex]\[
\binom{4}{0} x^{4-0} y^0
\][/tex]
Calculating the binomial coefficient [tex]\(\binom{4}{0}\)[/tex]:
[tex]\[
\binom{4}{0} = \frac{4!}{0! \cdot 4!} = 1
\][/tex]
So, the first term becomes:
[tex]\[
1 \cdot x^{4} \cdot y^0 = x^4
\][/tex]
Thus, the first term is:
[tex]\[
x^4
\][/tex]
### Part B: Determine the second term of the binomial expansion [tex]\((x + y)^4\)[/tex].
The second term corresponds to [tex]\(k = 1\)[/tex]. So, we take:
[tex]\[
\binom{4}{1} x^{4-1} y^1
\][/tex]
Calculating the binomial coefficient [tex]\(\binom{4}{1}\)[/tex]:
[tex]\[
\binom{4}{1} = \frac{4!}{1! \cdot 3!} = \frac{4 \cdot 3!}{1 \cdot 3!} = 4
\][/tex]
So, the second term becomes:
[tex]\[
4 \cdot x^3 \cdot y^1 = 4x^3 y
\][/tex]
Thus, the second term is:
[tex]\[
4x^3y
\][/tex]
