User:Aravind V R/Formulas

Linear Algebra
See User:Aravind V R/Formulas/Linear Algebra

GCD
Euclid's algorithm:
 * $$\gcd(a,0) = a$$
 * $$\gcd(a,b) = \gcd(b, a \,\mathrm{mod}\, b)$$,

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

Eisenstein's criterion
Suppose we have the following polynomial with integer coefficients.


 * $$Q=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$

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
 * $p^{2}$ does not divide $a_{0}$,

then $Q$ is irreducible over the rational numbers.

Rational root theorem
If a0 and an 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 an.