Sure, let's expand the expression [tex]\(\left(x + \frac{1}{x}\right)^2\)[/tex].
Step 1: Write down the expression:
[tex]\[
\left(x + \frac{1}{x}\right)^2
\][/tex]
Step 2: Recognize that this is a binomial squared, which follows the pattern:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = \frac{1}{x}\)[/tex].
Step 3: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the binomial expansion pattern:
[tex]\[
(x + \frac{1}{x})^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2
\][/tex]
Step 4: Simplify each term:
- The first term [tex]\(a^2\)[/tex] becomes [tex]\(x^2\)[/tex].
- The second term [tex]\(2ab\)[/tex] becomes [tex]\(2 \cdot x \cdot \frac{1}{x} = 2\)[/tex].
- The third term [tex]\(b^2\)[/tex] becomes [tex]\(\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}\)[/tex].
Step 5: Combine the simplified terms:
[tex]\[
x^2 + 2 + \frac{1}{x^2}
\][/tex]
Therefore, the expanded form of [tex]\(\left(x + \frac{1}{x}\right)^2\)[/tex] is:
[tex]\[
x^2 + 2 + \frac{1}{x^2}
\][/tex]
