Sure! Let's solve this step by step.
We're asked to find the coefficient of \(x^4\) in the expansion of \(\left(x + \frac{1}{2}\right)^4\).
1. Understand the Expansion:
Recall the binomial theorem, which states:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
In this case, \(a = x\), \(b = \frac{1}{2}\), and \(n = 4\).
2. Set Up the Relevant Terms:
We're interested in the term where the power of \(x\) is \(4\). Using the binomial theorem, this term is represented when the exponent of \(x\) is \(4\):
[tex]\[
\binom{4}{4} x^4 \left(\frac{1}{2}\right)^{4-4}
\][/tex]
3. Calculate the Coefficient:
[tex]\[
\binom{4}{4} x^4 \left(\frac{1}{2}\right)^{0} = \binom{4}{4} x^4 = 1 \cdot x^4
\][/tex]
Here, \(\binom{4}{4}\) is the binomial coefficient, which evaluates to \(1\).
So, the coefficient of [tex]\(x^4\)[/tex] in the expansion of [tex]\(\left(x + \frac{1}{2}\right)^4\)[/tex] is 1.
