Question

The coefficient of the highest power of \( x \) in the expansion of \[ \left( x + \sqrt{x^2 - 1} \right)^3 + \left( x - \sqrt{x^2 - 1} \right)^8 \text{ is:} \]

• A. \( 64 \)
• B. \( 128 \)
• C. \( 256 \)
• D. \( 512 \)

Hint:

In binomial expansions, the highest power of \( x \) in the expansion of \( (x + \text{term})^n \) is given by \( x^n \), and the coefficient is derived from the binomial expansion coefficients.

Answer 1

The Correct Option is C

We are tasked with finding the coefficient of the highest power of \( x \) in the expansion of: \[ \left( x + \sqrt{x^2 - 1} \right)^3 + \left( x - \sqrt{x^2 - 1} \right)^8 \] Step 1: Expand \( \left( x + \sqrt{x^2 - 1} \right)^3 \). Using the binomial theorem: \[ \left( x + \sqrt{x^2 - 1} \right)^3 = x^3 + 3x^2 \sqrt{x^2 - 1} + 3x \left( x^2 - 1 \right) + \left( x^2 - 1 \right)^{3/2} \] The highest power of \( x \) in this expansion is \( x^3 \). Step 2: Expand \( \left( x - \sqrt{x^2 - 1} \right)^8 \). Again, using the binomial theorem: \[ \left( x - \sqrt{x^2 - 1} \right)^8 = x^8 - 8x^7 \sqrt{x^2 - 1} + 28x^6 \left( x^2 - 1 \right) + \cdots \] The highest power of \( x \) here is \( x^8 \). Step 3: Combine the highest powers of \( x \) from both expansions: - From the first expansion: the highest power of \( x \) is \( x^3 \) - From the second expansion: the highest power of \( x \) is \( x^8 \) The highest power of \( x \) in the sum is \( x^8 \), and the coefficient of \( x^8 \) is 256. Thus, the answer is \( 256 \). % Final Answer \[ \boxed{256} \]