Binomial Formula and Coefficients

For $n$ positive integer $n=1,2,...$

$(x+y)^n=x^n+n x^{n-1}y + \dfrac{n(n-1)}{2!}x^{n-2}y^2+\dfrac{n(n-1)(n-2)}{3!}x^{n-3}y^3+...+y^n$

where $n!=1 \cdot 2 \cdot 3 ....\cdot (n-1) \cdot n$ and $0!=1$

Using the binomial coefficients, the above formula can be written as

$(x+y)^n=\dbinom{n}{0}x^n+ \dbinom{n}{1} x^{n-1}y + \dbinom{n}{2}x^{n-2}y^2+...+\dbinom{n}{k}x^{n-k}y^k+...+\dbinom{n}{n}y^n$

where

$\dbinom{n}{k} = \dfrac{n!}{k!(n-k)!}$

Examples

$\dbinom{n}{0} = \dfrac{n!}{0!(n-0)!}=\dfrac{n!}{1\cdot n!}=1$

$\dbinom{7}{5} = \dfrac{7!}{5!(7-5)!}=\dfrac{7 \cdot 6}{2\cdot 1}=21$

Use of Binomial Formula: Examples

1. $(x+y)^2=x^2+2 x y+y^2$
   
2. $(x+y)^3=x^3+3 x^2 y+3x y^2+y^3$
   
3. $(x-y)^2=x^2-2 x y+y^2$
   
4. $(x+y)^3=x^3-3 x^2 y+3x y^2-y^3$

Factoring Formulas

1. $x^2-y^2=(x-y)(x+y)$
   
2. $x^3-y^3=(x-y)(x^2+x y+y^2)$
   
3. $x^3+y^3=(x+y)(x^2-x y+y^2)$
   
4. $x^4-y^4=(x^2)^2-(y^2)^2=(x^2-y^2)(x^2+y^2)=(x-y)(x+y)(x^2+y^2)$