When expanded, which expression has a non-zero constant term? A - HSC - SSCE Mathematics Extension 1 - Question 9 - 2017 - Paper 1

Using the binomial theorem, the general term is:

$T_k = \binom{7}{k} (x^2)^{7-k} \left( \frac{1}{x^3} \right)^{k} = \binom{7}{k} x^{14 - 5k}$

Setting the exponent of $x$ to zero:

$14 - 5k = 0 \Rightarrow k = \frac{14}{5}$

Again, $k$ is not an integer, so there is no constant term.

The general term is:

$T_k = \binom{7}{k} (x^3)^{7-k} \left( \frac{1}{x^4} \right)^{k} = \binom{7}{k} x^{21 - 7k}$

Setting the exponent of $x$ to zero:

$21 - 7k = 0 \Rightarrow k = 3$

Since $k$ is an integer, this expression has a constant term.

The general term is:

$T_k = \binom{7}{k} (x^4)^{7-k} \left( \frac{1}{x^5} \right)^{k} = \binom{7}{k} x^{28 - 9k}$

Setting the exponent of $x$ to zero:

$28 - 9k = 0 \Rightarrow k = \frac{28}{9}$

Once more, $k$ is not an integer, so this expression does not have a constant term.