Binomial Theorem
Grade 11
Question:
<p>The expression <span class="math-tex">\(\left(x+\sqrt{x^{2}-1}\right)^{5}+\left(x-\sqrt{x^{2}-1}\right)^{5}\)</span> is a polynomial of degree</p>
<p style="display:inline">10</p>
<p style="display:inline">20</p>
<p style="display:inline">5</p>
<p style="display:inline">6</p>
Step-by-Step Solution
Key Concept: When expanding (a+b)^n + (a-b)^n using the Binomial Theorem, only even-powered terms of b survive due to cancellation of odd-powered terms. Here, the irrational parts ±√(x²-1) cancel in odd-indexed binomial coefficients, leaving only terms with even powers of √(x²-1), which become polynomial terms in x.
<p><span class="math-tex">$\left(x+\sqrt{x^{2}-1}\right)^{5}+\left(x-\sqrt{x^{2}-1}\right)^{5}$</span><br />
= 2[x<sup>5</sup> + <sup>5</sup>C<sub>2</sub>x<sup>3</sup> <span class="math-tex">$\left(\sqrt{x^{2}-1}\right)^{2}$</span> + <sup>5</sup>C<sub>4</sub>x<sup>1</sup> <span class="math-tex">$\left(\sqrt{x^{2}-1}\right)^{4}$</span>]<br />
= 2x<sup>5</sup> + 2 <span class="math-tex">$\cdot$</span> <sup>5</sup>C<sub>2</sub>x<sup>3</sup> (x<sup>2</sup> - 1) + 2 <span class="math-tex">$\cdot$</span> <sup>5</sup>C<sub>4</sub> x(x<sup>2</sup> - 1)<sup>2</sup><br />
which is a polynomial of degree 5.</p>
Correct Answer: C