Question

The coefficient of the highest power of \(x\) in the expansion of \((x + \sqrt{x^2 - 1})^8 + (x - \sqrt{x^2 - 1})^8\) is:

• A. 64
• B. 128
• C. 256
• D. 512

Hint:

Always consider trigonometric and hyperbolic identities when dealing with complex binomial expressions to simplify calculation.

Answer 1

The Correct Option is C

This problem involves simplifying two terms raised to the eighth power. The terms inside the parentheses are structured such that they effectively represent hyperbolic cosine functions. 
Step 1: Identify the simplification strategy
\[{Since } (x+\sqrt{x^2-1})^8 + (x-\sqrt{x^2-1})^8\] 
\[= 2\{{}^8C_0 x^8 + {}^8C_2 x^6 (x^2-1) + {}^8C_4 x^4 (x^2-1)^2 + {}^8C_6 x^2 (x^2-1)^3 + {}^8C_8 x^0 (x^2-1)^4 \}\] 
\[{So coefficient of highest power of } x \]
\[= 2\{{}^8C_0 + {}^8C_2 + {}^8C_4 + {}^8C_6 + {}^8C_8 \}\] 
\[= (1+1)^8 + (1-1)^8 = 2^8 = 256\] This corresponds to the highest power and matches option (C).

Answer 2

To find the coefficient of the highest power of \(x\) in the expansion of \((x + \sqrt{x^2 - 1})^8 + (x - \sqrt{x^2 - 1})^8\), observe the following:

Let \(a = x + \sqrt{x^2 - 1}\) and \(b = x - \sqrt{x^2 - 1}\).
The expression is \(a^8 + b^8\).

For large \(x\), analyze the behavior of \(a\) and \(b\):

• \(a = x + \sqrt{x^2 - 1} \approx x+x = 2x\)
• \(b = x - \sqrt{x^2 - 1} \approx x-x = 0\)

Thus, \(b^8\) becomes negligible for high powers of \(x\), and \(a^8\) dominates.

The highest power of \(x\) in \(a^8\) is \( (2x)^8 = 256x^8\).

Therefore, combining both expansions, since \(b^8\) contributes virtually nothing, gives the coefficient:
256.