Theory of Equations – 18UMTC12
Mrs.P.Kalaiselvi, M.Sc.,M.A.,
Ms.S.Swathi Sundari, M.Sc.,M.Phil.,

Preliminaries - Polynomials

Preliminaries
Solving an equations by using
various methods
Algebraic Long Division Method
Synthetic Division Method
Completing The Square Method
Factorization Method
Special Quadratic Formula (only for solving
quadratic equation)

Preliminaries – Solving an equation by using
Algebraic Long Division Method

Preliminaries – Solving an equation by using

Preliminaries – Solving an equation by using
Factorization Method

Preliminaries – Solving an equation by using

Preliminaries – Solving an equation by using
Special Quadratic Formula

Preliminaries – Solving an equation by using
Special Quadratic Formula

RESULTS
Remainder Theorem : If f(x) is a polynomial, then f(a) is
the remainder when f(x) is divided by x-a.
If f(a) and f(b) are of different signs, then at least one
root of the equation f(x)=0 must lie between a and b.
If f(x)=0 is an equation of odd degree, it has at least one
real root whose sign is opposite to that of the last term.
If f(x)=0 is an equation of even degree and the absolute
term is negative, equation has at least one positive root
and at least one negative root.
Every equation f(x)=0 of the degree has n roots and
no more.
In an equation with real (rational) coefficients, imaginary
(irrational) roots occur in pairs.

Relation Between Roots and the Coefficients

Transformations of Equations
Reciprocal Equations : An equation in which the reciprocal of every
root of the equation is also its root is called a reciprocal equation. (Or)
An equation, which remains unchanged when x is replaced by 1/x is
called a reciprocal equation.
Note: In such an equation, the coefficients from one end are equal to
the coefficients from the other end (or) Equal in magnitude and
opposite in sign.
Remark:
 When an odd degree equation,
• If the coefficients have like signs, then -1 is a root.
• If the coefficients of the terms equidistant from the first and last
have opposite signs, then +1 is a root.
 The degree is even and the coefficients of the terms equidistant
from the first and last are equal and have the same sign.

Transformations of Equations
 If are the roots of f(x)=0, the equation
i. Whose roots are is f(1/x) =0.
ii. Whose roots are is f(x/k) =0.
iii.Whose roots are is f(x+h) =0.
iv.Whose roots are is f(x-h) =0.
v. Whose roots are is f(√x) =0.
 In an reciprocal equation, increasing by h the roots of the
equation is the same as diminishing the roots by –h.