To simplify the expression [tex]\(\left(x + \frac{1}{x}\right)^2\)[/tex], we can expand it using the formula for the square of a binomial. Here's the detailed, step-by-step solution:
1. Write the expression:
[tex]\[
\left(x + \frac{1}{x}\right)^2
\][/tex]
2. Expand the square using the binomial theorem:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = \frac{1}{x}\)[/tex]. Applying the binomial theorem, we get:
[tex]\[
\left(x + \frac{1}{x}\right)^2 = x^2 + 2 \left(x \cdot \frac{1}{x}\right) + \left(\frac{1}{x}\right)^2
\][/tex]
3. Simplify each term:
- The first term is [tex]\(x^2\)[/tex].
- The second term is [tex]\(2 \left(x \cdot \frac{1}{x}\right)\)[/tex]. Since [tex]\(x \cdot \frac{1}{x} = 1\)[/tex], the second term simplifies to [tex]\(2 \cdot 1 = 2\)[/tex].
- The third term is [tex]\(\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}\)[/tex].
4. Combine all the terms:
[tex]\[
x^2 + 2 + \frac{1}{x^2}
\][/tex]
So, the expanded form of the expression [tex]\(\left(x + \frac{1}{x}\right)^2\)[/tex] is:
[tex]\[
x^2 + 2 + \frac{1}{x^2}
\][/tex]
However, we can represent this in a different form by combining the terms under a single fraction, if desired:
[tex]\[
\frac{x^4 + 2x^2 + 1}{x^2}
\][/tex]
This expression can be factored further as:
[tex]\[
\frac{(x^2 + 1)^2}{x^2}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\left(x + \frac{1}{x}\right)^2\)[/tex] is:
[tex]\[
\frac{(x^2 + 1)^2}{x^2}
\][/tex]
