Theory of Equation

Transformations of Equations

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
RemainderTheorem:Iff(x)isapolynomial,thenf(a)is
theremainderwhenf(x)isdividedbyx-a.
Iff(a)andf(b)areofdifferentsigns,thenatleastone
rootoftheequationf(x)=0mustliebetweenaandb.
Iff(x)=0isanequationofodddegree,ithasatleastone
realrootwhosesignisoppositetothatofthelastterm.
Iff(x)=0isanequationofevendegreeandtheabsolute
termisnegative,equationhasatleastonepositiveroot
andatleastonenegativeroot.
Everyequationf(x)=0ofthe degreehasnrootsand
nomore.
Inanequationwithreal(rational)coefficients,imaginary
(irrational)rootsoccurinpairs.

12. Relation Between Roots and the Coefficients

13. Transformationsof Equations
ReciprocalEquations:Anequationinwhichthereciprocalofevery
rootoftheequationisalsoitsrootiscalledareciprocalequation.(Or)
Anequation,whichremainsunchangedwhenxisreplacedby1/xis
calledareciprocalequation.
Note:Insuchanequation,thecoefficientsfromoneendareequalto
thecoefficientsfromtheotherend(or)Equalinmagnitudeand
oppositeinsign.
Remark:
Whenanodddegreeequation,
•Ifthecoefficientshavelikesigns,then-1isaroot.
•Ifthecoefficientsofthetermsequidistantfromthefirstandlast
haveoppositesigns,then+1isaroot.
Thedegreeisevenandthecoefficientsofthetermsequidistant
fromthefirstandlastareequalandhavethesamesign.

14. Transformationsof Equations
If aretherootsoff(x)=0,theequation
i.Whoserootsare isf(1/x)=0.
ii.Whoserootsare isf(x/k)=0.
iii.Whoserootsare isf(x+h)=0.
iv.Whoserootsare isf(x-h)=0.
v.Whoserootsare isf(√x)=0.
Inanreciprocalequation,increasingbyhtherootsofthe
equationisthesameasdiminishingtherootsby–h.