Question 12 Marks
Find the coefficient of $x$ in the expansion of $(1 - 3x + 7x^2) (1 - x)^{16}.$
Answer
View full question & answer→$(1-3\text{x}+1\text{x}^2)(1-\text{x})^{16}$
$=(1-3\text{x}+7\text{x}^2)(\ ^{16}\text{C}_0-\ ^{16}\text{C}_1\text{x}^1+\ ^{16}\text{C}_2\text{x}^2+\ ...+\ ^{16}\text{C}_{16}\text{x}^{16})$
$=(1-3\text{x}+7\text{x}^2)(1-16\text{x}+120\text{x}^2+\ ...)$
$\therefore$ Coefficient of $\text{x}=-16-3=-19$
$=(1-3\text{x}+7\text{x}^2)(\ ^{16}\text{C}_0-\ ^{16}\text{C}_1\text{x}^1+\ ^{16}\text{C}_2\text{x}^2+\ ...+\ ^{16}\text{C}_{16}\text{x}^{16})$
$=(1-3\text{x}+7\text{x}^2)(1-16\text{x}+120\text{x}^2+\ ...)$
$\therefore$ Coefficient of $\text{x}=-16-3=-19$