Write the general term in the expansion of (x2 – yx)12, x ≠ 0

Question:

Write the general term in the expansion of $\left(x^{2}-y x\right)^{12}, x \neq 0$

Solution:

It is known that the general term $T_{r+1}$ \{which is the $(r+1)^{\text {th }}$ term $\}$ in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$.

Thus, the general term in the expansion of $\left(x^{2}-y x\right)^{12}$ is

$\mathrm{T}_{\mathrm{r}+1}={ }^{12} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{2}\right)^{12-\mathrm{t}}(-\mathrm{yx})^{\mathrm{r}}=(-1)^{\mathrm{r}}{ }^{12} \mathrm{C}_{\mathrm{r}} \cdot \mathrm{x}^{24-2 \mathrm{r}} \cdot \mathrm{y}^{\mathrm{r}} \cdot \mathrm{x}^{\mathrm{r}}=(-1)^{\mathrm{r}}{ }^{12} \mathrm{C}_{\mathrm{r}} \cdot \mathrm{x}^{24-\mathrm{r}} \cdot \mathrm{y}^{\mathrm{r}}$

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