Basic Algebraic formulae with summary of formulae

(a + b)^2 = a^2 + 2*a*b + b^2

(a - b)^2 = a^2 - 2*a*b + b^2

(a + b + c)^2 = a^2 + b^2 + c^2 + 2*(ab + bc + ca)

(a + b)^3 = a^3 + b^3 + 3*a*b*(a + b)

(a - b)^3 = a^3 - b^3 - 3*a*b*(a - b)

(a + b + c)^3 = a^3 + b^3 + c^3 + 3*(a + b)*(b + c)*(c + a)

(a + b)^2 = (a - b)^2 + 4*a*b

(a - b)^2 = (a + b)^2 – 4*a*b

a^2 – b^2 = (a + b)(a - b)

a^3 + b^3 = (a + b)(a^2 + b^2 - ab)

a^3 – b^3 = (a - b)(a^2 + b^2 + ab)

a^3 + b^3 +c^3 – 3*a*b*c = (a + b + c)*(a^2 + b^2 + c^2 - a*b - b*c - c*a)

       a^n + b^n is divisible by (a + b), when n is odd.

       a^n + b^n is not divisible by (a + b), when n is even.

       a^n + b^n is not divisible by (a - b).

       a^n - b^n is always divisible by (a - b).

       a^n - b^n is not divisible by (a + b), when n is odd.

       a^n - b^n is divisible by (a + b), when n is even. 

 

Descartes rule of Signs

For any polynomial f(x) = 0,

The maximum number of positive real roots of a equation is the number of changes of signs from positive to negative and negative to positive in f(x).

The maximum number of negative real roots of a equation is the number of changes of signs from positive to negative and negative to positive in f(-x).

Remaining roots are imaginary. 

f(x) = a*x^3 + b*x^2 – c*x + d = 0

         +         +            -         +

       Thus, we have two sign changes. Hence there are at most two positive roots.

f(-x) = -a*x^3 + b*x^2 + c*x + d = 0

           -           +            +        +

       Thus, we have one sign change. Hence there is at most one negative root.

Polynomial equations

A polynomial equation is an equation of degree n where n is greater than 2. 

Sum of roots taking one at a time = [coefficient of x^(n-1)]/[coefficient of x^n] = an-1/an.
Sum of roots taking r at a time = [(-1)^r]*[coefficient of x^(n-r)]/[coefficient of x^n] = [(-1)^r]*an-r/an

Product of all roots = (-1)^n * [coefficient of x^0]/[coefficient of x^n] = (-1)^n * a1/an.