Binomial Expansion
Forn= 1, 2 , 3 ,...an integer, then(x+y)n=xn+nxn−^1 y+n(n−1)
2!xn−^2 y^2 +n(n−1)(n−2)
3!xn−^3 y^3 +···+ynwheren!is read n factorial and is defined
n! =n(n−1)(n−2)··· 3 · 2 · 1 and0! = 1by definition.Binomial Coefficients
The binomial coefficients can also be defined by the expression
(
n
k)
= n!
k!(n−k)!wheren! =n(n−1)(n−2)··· 3 · 2 · 1where forn = 1, 2 , 3 ,...is an integer. The binomial expansion has the alternative
representation
(x+y)n=(
n
0)
xn+(
n
1)
xn−^1 y+(
n
2)
xn−^2 y^2 +(
n
3)
xn−^3 y^3 ···+(
n
n)
ynLaws of Exponents
Letsandtdenote real numbers and letmandndenote positive integers.For nonzero values ofxandy
x^0 = 1, x 6 = 0
xsxt=xs+t
xs
xt=xs−t(xs)t=xst
(xy)s=xsys
x−s=^1
xsx^1 /n=n√
x
xm/n=n√
xm
(
x
y) 1 /n
=x1 /n
y^1 /n=√nx
√nyLaws of Logarithms
Ifx=byandb 6 = 0, then one can writey= logbx, wherey is called the logarithmofxto the baseb. ForP > 0 andQ > 0 , logarithms satisfy the following properties
logb(PQ) = logbP+ logbQlogbP
Q= logbP−logbQ
logbQP=PlogbQ