Some Standard Expansions - Practice Questions & MCQ

We know the binomial expansion,

$
(\mathrm{x}+\mathrm{y})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_0 \mathrm{x}^{\mathrm{n}}+{ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}^{\mathrm{n}-1} \mathrm{y}+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}^{\mathrm{n}-2} \mathrm{y}^2+\cdots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{y}^{\mathrm{n}}
$

i.e. $\quad(x+y)^n=\sum_{r=0}^n{ }^n C_r x^{n-r} y^r$
1. Replace ' $y$ ' with ' $-y$ ' in the binomial expansion, we get

$
\begin{aligned}
& (x-y)^n={ }^n C_0 x^n-{ }^n C_1 x^{n-1} y+{ }^n C_2 x^{\mathrm{n}-2} y^2-\cdots+(-1)^{\mathrm{r} n} C_r x^{\mathrm{n}-\mathrm{r}} y^{\mathrm{r}}+\cdots+(-1)^{\mathrm{n}{ }^n} C_n y^{\mathrm{n}} \\
& \text { or } \quad(\mathrm{x}-\mathrm{y})^{\mathrm{n}}=\sum_{\mathrm{r}=0}^{\mathrm{n}}(-1)^{\mathrm{r} n} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{n}-\mathrm{r}} y^{\mathrm{r}}
\end{aligned}
$

2. In the binomial expansion, $(x+y)^n$ replace ' $x$ ' by 1 and ' $y$ ' by $x$

$
\begin{aligned}
& (1+\mathrm{x})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_0 \mathrm{x}^0+{ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}^1+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}^2+\cdots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}+\cdots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \\
& \text { or } \quad(1+\mathrm{x})^{\mathrm{n}}=\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}
\end{aligned}
$

3. In the binomial expansion, $(x+y)^n$ replace ' $x$ ' by ' 1 ' and ' $y$ ' by ' $-x$ '

$
\begin{aligned}
& (1-\mathrm{x})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_0 \mathrm{x}^0-{ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}^1+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}^2-\cdots+(-1)^{\mathrm{r}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}+\cdots+(-1)^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \\
& \text { or } \quad(1-x)^n=\sum_{r=0}^n(-1)^r{ }^n C_r x^r
\end{aligned}
$

4. Addition: $(x+y)^n+(x-y)^n$

$
(x+y)^n+(x-y)^n=2\left[{ }^n C_0 x^n y^0+{ }^n C_2 x^{n-2} y^2+{ }^n C_4 x^{n-4} y^4+\ldots .\right]
$
If ' $n$ ' is odd then number of terms is $\frac{n+1}{2}$
If $n^{\prime}$ is even then number of terms is $\frac{n^2}{2}+1$
5. Subtraction: $(x+y)^n-(x-y)^n$

$
(x+y)^n-(x-y)^n=2\left[{ }^n C_1 x^{n-1} y^1+{ }^n C_3 x^{n-3} y^3+{ }^n C_5 x^{n-5} y^5+\ldots \ldots\right]
$
If $n$ is odd, then the number of terms is $\frac{n+1}{2}$
If $n$ is even, then the number of terms is $\frac{n}{2}$