Expand the following by binomial theorem: (x² + 2/x)

x ≠ 0

To expand the expression \((x^2 + \frac{2}{x})\) using the binomial theorem, we first recognize that it can be treated as a binomial raised to a power. In this case, we will expand it as \((x^2 + \frac{2}{x})^1\), since there is no exponent indicated. The binomial theorem states that:

Binomial Theorem Overview

The binomial theorem provides a formula for expanding expressions of the form \((a + b)^n\), where:

• a and b are any numbers or expressions.
• n is a non-negative integer.

Applying the Theorem

For our expression, we can directly apply the theorem:

• a = x^2
• b = \frac{2}{x}
• n = 1

Using the binomial expansion formula, we have:

\((a + b)^n = a^n + \binom{n}{1} a^{n-1} b + b^n\)

Calculating the Terms

Now we can calculate each term:

• The first term: \( (x^2)^1 = x^2 \)
• The second term: \( \binom{1}{1} (x^2)^0 \left(\frac{2}{x}\right)^1 = 1 \cdot 1 \cdot \frac{2}{x} = \frac{2}{x} \)
• The third term: \( \left(\frac{2}{x}\right)^0 = 1 \)

Final Expansion

Putting it all together, we get:

\((x^2 + \frac{2}{x})^1 = x^2 + \frac{2}{x} + 1\)

Thus, the expanded form of \((x^2 + \frac{2}{x})\) is:

x^2 + \frac{2}{x} + 1