User:Aravind V R/Formulas

Linear Algebra

See User:Aravind V R/Formulas/Linear Algebra

Algebra

GCD

Euclid's algorithm:

${\displaystyle \gcd(a,0)=a}$ 

${\displaystyle \gcd(a,b)=\gcd(b,a\,\mathrm {mod} \,b)}$ ,

Properties:

1. ${\displaystyle \gcd(n^{a}-1,n^{b}-1)=n^{\gcd(a,b)}-1\,}$ 
2. Bézout's identity: If d=gcd(a,b) then d can be written as ${\displaystyle d=ap+qb\,}$  where p and q are integers

Eisenstein's criterion

Suppose we have the following polynomial with integer coefficients.

${\displaystyle Q=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$ 

If there exists a prime number p such that the following three conditions all apply:

• p divides each a_i for i ≠ n,
• p does not divide a_n, and
• p² does not divide a₀,

then Q is irreducible over the rational numbers.

Rational root theorem

If a₀ and a_n are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies

• p is an integer factor of the constant term a₀, and
• q is an integer factor of the leading coefficient a_n.

Probability

Main page: User:Aravind V R/Formulas/Probability

Information theory

Main page: User:Aravind V R/Formulas/Information theory