This document discusses various methods for solving polynomial equations, including algebraic long division, synthetic division, completing the square, factorization, and the quadratic formula. It also covers key results about polynomial equations like the remainder theorem, relationships between roots and coefficients, and transformations of equations. Specifically, it defines reciprocal equations as those where taking the reciprocal of every root also yields a root, and notes properties of odd and even degree reciprocal equations. It also outlines how transformations like shifting (f(x+h)), diminishing (f(x-h)), and scaling (f(x/k)) affect the roots of an equation.

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

2. Preliminaries - Polynomials

3. 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)

4. Preliminaries – Solving an equation by using
Algebraic Long Division Method

5. Preliminaries – Solving an equation by using

6. Preliminaries – Solving an equation by using
Factorization Method

7. Preliminaries – Solving an equation by using

9. Preliminaries – Solving an equation by using
Special Quadratic Formula

10. Preliminaries – Solving an equation by using
Special Quadratic Formula

11. 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.

12. Relation Between Roots and the Coefficients

13. 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.

14. 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.