Types of Equations

Types of Equations

Let us learn one of the most important topics of Equations.i.e. the Types of Equations.
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.

Points to Remember about Equations:

1. Every algebraic equation of degree n always has exactly n roots.

2. Suppose an algebraic equation with real coefficients has a complex root a+ib, then this equation has also the complex conjugate root a - ib with the multiplities being the same.

3. Descartes rule of signs. In a polynomial with real coefficients number of positive roots cannot exceed the number of variation of sign of the coefficient in f(x) and number of negative roots cannot exceed the number of variations of sign of the coefficients in f(-x).

4. If f(x) is continuous on [a, b] and f (a), f (b) are opposite signs, then f(x) = 0 has at least one real root lying in the interval (a, b).

We can factorize the algebraic expression by using factor theorem and hence find roots in some cases. This can be done only on trial basis.

Hope you like the above example of Types of Equations.Please leave your comments, if you have any doubts.