Binomial Theorem
Grade 11
Question:
<p>If f(x) = 1 - x + x<sup>2</sup> - x<sup>3</sup> + ... - x<sup>15</sup> + x<sup>16</sup> - x<sup>17</sup>, then the coefficient of x<sup>2</sup> in f(x - 1) is</p>
<p style="display:inline">816</p>
<p style="display:inline">822</p>
<p style="display:inline">828</p>
<p style="display:inline">848</p>
Step-by-Step Solution
Key Concept: Represent the alternating polynomial as a geometric progression sum to simplify the substitution of (x-1) before applying the binomial theorem.
<p>f(x) = 1 - x + x<sup>2</sup> - x<sup>3</sup> + ... -x<sup>15</sup> + x<sup>16</sup> - x<sup>17</sup><br />
<span class="math-tex">$=\frac{1-(-x)^{18}}{1-(-x)}$</span><br />
<span class="math-tex">$=\frac{1-x^{18}}{1+x}$</span><br />
<span class="math-tex">$\therefore$</span> f(x - 1) = <span class="math-tex">$\frac{1-(x-1)^{18}}{1+x-1}$</span><br />
<span class="math-tex">$=\frac{1-(x-1)^{18}}{x}$</span><br />
<span class="math-tex">$=\frac{1}{x}-\frac{1}{x}$</span> (x - 1)<sup>18</sup><br />
<span class="math-tex">$=\frac{1}{x}-\frac{1}{x}$</span> (1 - x)<sup>18</sup><br />
<span class="math-tex">$=\frac{1}{x}-\frac{1}{x}$</span> (<sup>18</sup>C<sub>0</sub> - <sup>18</sup>C<sub>1</sub>x + <sup>18</sup>C<sub>2</sub>x<sup>2</sup> - ... + <sup>18</sup>C<sub>18</sub>x<sup>18</sup>)<br />
= x<sup>-1</sup> - <sup>18</sup>C<sub>0</sub>x<sup>-1</sup> + <sup>18</sup>C<sub>1</sub>x<sup>0</sup> - <sup>18</sup>C<sub>2</sub>x + <sup>18</sup>C<sub>3</sub>x<sup>2</sup> - ... - <sup>18</sup>C<sub>18</sub>x<sup>17</sup><br />
<span class="math-tex">$\therefore$</span> coefficient of x<sup>2</sup> = <sup>18</sup>C<sub>3</sub> = 816</p>
Correct Answer: A