WAEC Further Mathematics 2012 Paper

Concept: The constant term occurs when the powers of $x$ cancel completely in the binomial expansion.Explanation:For the expansion of $(a+b)^n$, the general term is$$T_r=\binom{n}{r}a^{n-r}b^r.$$Here, $n=8$, $a=2x$, and $b=-\frac{3}{x}$, so$$T_r=\binom{8}{r}(2x)^{8-r}\left(-\frac{3}{x}\right)^r.$$The constant term has no $x$, so the exponent of $x$ must be zero.The power of $x$ in $(2x)^{8-r}$ is $8-r$.The power of $x$ in $\left(-\frac{3}{x}\right)^r$ is $-r$.Thus, $8-r-r=0 \Rightarrow 8-2r=0 \Rightarrow r=4$.Substitute $r=4$ to get$$\binom{8}{4}(2)^{4}(-3)^4 = 70 \times 16 \times 81 = 90720.$$Since $(-3)^4$ is positive, the constant term is positive.Answer:The constant term is $90720$.Correct option: A.