Expansions

 In algebra we come across certain products very frequently. For e.g., (a + b)², (a + b)³ (a + b + c)² etc. These are nothing but products of binomials or trinomials. We derive the formulae for these products and apply them whenever necessary. These expansions help us to avoid elaborate multiplications and make calculations quicker.

Equations

Fundamentals of Equations

Algebraic and transcendental equations;

If f(x) is a polynomial in x, then f(x) =0 is an algebraic equation.

Example; x⁷ + 5x - 2=0.

If f(x) contains algebraic and non algebraic functions namely exponential, logarithmic, trigonometric and inverse trigonometric functions then f(x) =0 is called transcendental equation.

Example; x + log x + sin x=0.

Transcendental equations may have no root, exactly one root or more than one root.

Algebraic equations:

Fundamental theorem on Algebra.

Every algebraic equation of degree n ≥ 1 has a root real or complex.

Expansions

     We shall discuss expansions of binomials, trinomials, Formulae

     1. a² + b² = (a + b)² - 2ab

     2. a² + b² = (a - b)² + 2ab

     3. (a + b)² = (a - b)² + 4ab

Summary

     (a + b) (a - b) = a² - b²

     (x + a) (x + b) = x² + (a + b)x + ab.

     (a + b)³ = a³ + b³ + 3ab (a + b) = a³ + 3a²b + 3ab² + b³.