Question

If the coefficient of $x$ in the expansion of $(1+ax{)}^8(1+3x{)}^4-(1+x{)}^3(1+2x{)}^4$ is zero, then $a$ equals to

• A. $\frac{1}{4}$
• B. $-\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $-\frac{1}{8}$

Answer 1

The correct option is D $-\frac{1}{8}$

Given expression is $(1+ax){}^8(1+3x){}^4-(1+x{)}^3(1+2x{)}^4$

Therefore the coefficient of $x$ is equal to

$8a+4\left(3\right)-\left(3\right(1)+4(2\left)\right)$

$=8a+12-3-8$

$=8a+4-3$

$=8a+1$

Since a coefficient of $x$ in the expansion is zero

$8a+1=0$

$a=\frac{-1}{8}$