Polynomial. Nature of roots. Types of quadratic equation. Relations between roots and coefficients. Solution of quadratic equation

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1 Qudrti Equtios d Iequtios Polyomil Algeri epressio otiig my terms of the form, eig o-egtive iteger is lled polyomil ie, f ( , where is vrile,,,, re ostts d Emple : , () Rel polyomil ( f + is lled rel polyomil of rel vrile with rel oeffiiets Emple: + 5, + et re rel polyomils () Comple polyomil f( is lled omple polyomil of omple vrile with omple oeffiiets Emple: ( + i) + (5i ), 5i + ( + i) + et re omple polyomils () Degree of polyomil : Highest power of vrile i polyomil is lled degree of polyomil Emple: f ( is degree polyomil f ( is degree polyomil A polyomil of seod degree is geerlly lled qudrti polyomil Polyomils of degree d re ow s ui d iqudrti polyomils respetively () Polyomil equtio : If f( is polyomil, rel or omple, the f( is lled polyomil equtio Types of qudrti equtio A equtio i whih the highest power of the uow qutity is two is lled qudrti equtio Qudrti equtios re of two types : Purely qudrti +, where, C d, Tle : 5 Adfeted qudrti + +, where,, C d, Roots of qudrti equtio : The vlues of vrile whih stisfy the qudrti equtio is lled roots of qudrti equtio Solutio of qudrti equtio () Ftoriztio method Let + + ( )( β) The d β will stisfy the give equtio Hee, ftorize the equtio d equtig eh ftor to zero gives roots of the equtio Emple : + ( )( + ) ;, / () Sri Dhrhry method : By ompletig the perfet squre s Addig d sutrtig, + ± whih gives, Hee the qudrti equtio + + ( ) hs two + roots, give y, β Every qudrti equtio hs two d oly two roots Nture of roots I qudrti equtio + +, let us suppose tht,, re rel d The followig is true out the ture of its roots () The equtio hs rel d distit roots if d oly if D > () The equtio hs rel d oiidet (equl) roots if d oly if D () The equtio hs omple roots of the form ± i β,, β R if d oly if D < () The equtio hs rtiol roots if d oly if,, Q (the set of rtiol umers) d D is perfet squre (of rtiol umer) (5) The equtio hs (uequl) irrtiol (surd form) roots if d oly if D > d ot perfet squre eve if, d re rtiol I this se if irrtiol root, the p + q, p, q rtiol is p q is lso root (,, eig rtiol) (6) + iβ ( β d, β R ) is root if d oly if its ojugte iβ is root, tht is omple roots our i pirs i qudrti equtio I se the equtio is stisfied y more th two omple umers, the it redues to idetity + +, ie, Reltios etwee roots d oeffiiets () Reltio etwee roots d oeffiiets of qudrti equtio : If d β re the roots of qudrti equtio + +, ( ) the Coeffiiet of Sum of roots S + β Coeffiiet of Costt term Produt of roots P β Coeffiie t of () Formtio of equtio with give roots : A qudrti equtio whose roots re d β is give y ( )( β) ( + β) + β ie (sum of roots) + (produt of roots) S + P () Symmetri futio of the roots : A futio of d β is sid to e symmetri futio, if it remis uhged whe d β re iterhged

2 Qudrti Equtios d Iequtios For emple, + β + β is symmetri futio of d β wheres β + β is ot symmetri futio of d β I order to fid the vlue of symmetri futio of d β, epress the give futio i terms of + β d β The followig results my e useful (i) + β ( + β) β (ii) + β ( + β) β ( + β) (iii) + β ( + β )( + β) β( + β ) 5 5 (iv) + β ( + β )( + β ) β ( + β) (v) β ( + β) β (vi) β ( + β)( β) (vii) β ( β)[( + β) β ] (viii) β ( + β)( β)( + β ) Higher degree equtios The equtio p (i) ( Where the oeffiiets,,, R (or C) d is lled equtio of th degree, whih hs etly roots,,, C, the we write p ( )( )( ) ( { ( Σ ) + ( Σ ) + ( ) } Comprig (i) d (ii), Σ Σ + + d so o d ( ) (ii) Cui equtio : Whe, the equtio is ui of the form d, d we hve i this se + β + γ ; d β + βγ + γ ; βγ Biqudrti equtio : If, β, γ, δ re roots of the iqudrti equtio d + e, the σ σ σ + β + γ + δ β + γ + δ + βγ + βδ + γδ βγ + βδ + γδ + βγδ d e σ βγδ Formtio of polyomil equtio from give roots : If,,, re the roots of polyomil equtio of degree, the the equtio is + + σ + σ σ ( ) σ where σ r r Cui equtio : If, β, γ re the roots of ui equtio, the the equtio is σ + σ σ or ( + β + γ ) + ( β + γ + βγ ) βγ Biqudrti Equtio : If, β, γ, δ re the roots of iqudrti equtio, the the equtio is or σ + σ σ + σ ( + β + γ + δ ) + ( β + γ + δ + βγ + βδ + γδ ) ( βγ + βδ + γδ + βγδ) + βγδ Coditio for ommo roots () Oly oe root is ommo : Let e the ommo root of qudrti equtios + + d , + + By Crmmer s rule : or, The oditio for oly oe root ommo is ( ) ( )( ) () Both roots re ommo: The required oditio is Properties of qudrti equtio () If f() d f() re of opposite sigs the t lest oe or i geerl odd umer of roots of the equtio f ( lie etwee d () If f ( ) f( ) the there eists poit etwee d suh tht f ( ), < < () If is root of the equtio f ( the the polyomil f ( is etly divisile y ( ), the ( ) is ftor of f ( () If the roots of the qudrti equtios + + d + + re i the sme rtio i e the β β / / The qudrti epressio () Let f ( + +,,, R, > e qudrti epressio Sie, f ( + (i) The followig is true from equtio (i)

3 Qudrti Equtios d Iequtios (i) f ( > ( < ) for ll vlues of R if d oly if > ( < ) d D < (ii) f ( ( ) if d oly if > ( < ) d D I this se ( D ), f ( if d oly if (iii) If D > d > (<), the < ( > ), f ( > ( < ),, for lyig etwee the roots of f( for ot lyig etwee the roots of f( for eh of the roots of f( (iv) If >,( < ), the f ( hs miimum (mimum) vlue t d this vlue is give y [ f( ] mi (m () Sig of qudrti epressio : Let f ( + + or y + + Where,, R d, for some vlues of, f( my e positive, egtive or zero This gives the followig ses : (i) > d D <, so f ( > for ll R ie, f ( is positive for ll rel vlues of (ii) < d D <, so f ( < for ll R ie, f( is egtive for ll rel vlues of (iii) > d D, so f ( for ll R ie, f( is positive for ll rel vlues of eept t verte, where f ( (iv) < d D, so f ( for ll R ie f( is egtive for ll rel vlues of eept t verte, where f ( (v) > d D >, let f ( hve two rel roots d β ( < β), the f ( > for ll (, ) ( β, ) d f ( < for ll (, β) (vi) < d D >, let f ( hve two rel roots d β ( < β) The f ( < for ll (, ) ( β, ) d f ( > for ll (, β) () Grph of qudrti epressio We hve y + + f( y D + D Now, let y + Y d X + D y + + Y X X Y (i) The grph of the urve y f( is proli (ii) The is of prol is X or + ie (prllel to y-is) (iii) () If >, the the prol opes upwrd () If <, the the prol opes dowwrd >, D < -is -is <, D < (iv) Itersetio with is () Itersetio with -is : For is, y ± D + + For D >, prol uts -is i two rel d distit poits ie ± D For D,prol touhes -is i oe poit, / <, D > -is >, D > -is For D <, prol >, D does ot ut -is (ie imgiry vlue of >, D < () Itersetio with is y-is : For y is, y Wvy urve method ( ) ( ) Let f( ( ) ( ) ( ) (i) where,,, N d,,,, re fied turl umers stisfyig the oditio < < < < First we mr the umers,,,, o the rel is d the plus sig i the itervl of the right of the lrgest of these umers, ie o the right of If is eve the we put plus sig o the left of d if is odd the we put mius sig o the left of I the et itervl we put sig ordig to the followig rule : Whe pssig through the poit the polyomil f( hges sig if is odd umer d the polyomil f( hs sme sig if is eve umer The, we osider the et itervl d put sig i it usig the sme rule Thus, we osider ll the itervls The solutio of f ( > is the uio of ll itervls i whih we hve put the plus sig d the solutio of f ( < is the uio of ll itervls i whih we hve put the mius sig Positio of roots -is -is () If f ( is equtio d, re two rel umers suh tht f ( ) f( ) < hs t lest oe rel root or odd umer of rel roots etwee d I se f () d f () re of the sme sig, the either o rel root or eve umer of rel roots of f( lie etwee d () Every equtio of odd degree hs t lest oe rel root, whose sig is opposite to tht of its lst term, provided the <, D <, D < -is -is

4 Qudrti Equtios d Iequtios oeffiiet of the first term is +ve eg, + hs oe rel egtive root () Every equtio of eve degree whose lst term is ve d the oeffiiet of first term +ve hs t lest two rel roots, oe +ve d oe ve eg, hs t lest two rel roots, oe +ve d oe ve () If equtio hs oly oe hge of sig, it hs oe +ve root d o more (5) If ll the terms of equtio re +ve d the equtio ivolves o odd power of, the ll its roots re omple Desrte's rule of sigs The mimum umer of positive rel roots of polyomil equtio f ( is the umer of hges of sig from positive to egtive d egtive to positive i f( The mimum umer of egtive rel roots of polyomil equtio f ( is the umer of hges of sig from positive to egtive d egtive to positive i f( Rtiol lgeri iequtios () Vlues of rtiol epressio / for rel vlues of, where d re qudrti epressios : To fid the vlues ttied y rtiol epressio of the form for rel vlues of, the followig lgorithm will epli the proedure : Algorithm Step I: Equte the give rtiol epressio to y Step II: Oti qudrti equtio i y simplifyig the epressio i step I Step III: Oti the disrimit of the qudrti equtio i Step II Step IV: Put Disrimit d solve the iequtio for y The vlues of y so otied determies the set of vlues ttied y the give rtiol epressio () Solutio of rtiol lgeri iequtio: If d re polyomil i, the the iequtio >, <, d re ow s rtiol lgeri iequtios To solve these iequtios we use the sig method s eplied i the followig lgorithm Algorithm Step I: Oti d Step II: Ftorize d ito lier ftors Step III: Me the oeffiiet of positive i ll ftors Step IV: Oti ritil poits y equtig ll ftors to zero Step V: Plot the ritil poits o the umer lie If there re ritil poits, they divide the umer lie ito ( + ) regios Step VI: I the right most regio the epressio ers positive sig d i other regios the epressio ers positive d egtive sigs depedig o the epoets of the ftors () Lgrge s idetity If,,,, R the, + + )( + + ) ( + ) ( + ) + ( ) + ( ) ( Equtios whih e redued to lier, Qudrti d Biqudrti equtios Type I : A equtio of the form ( )( )( )( d) A, where of vrile ie, < < < d, d, e solved y hge ( ) + ( ) + ( ) + ( d) y ( d) y Type II : A equtio of the form ( )( )( )( d) A where d, e redued to olletio of two qudrti equtios y hge of vrile y + Type III : A equtio of the form ( ) + ( ) A lso e solved y hge of vrile, ie, mig sustitutio ( ) + ( ) y Some importt result () For the qudrti equtio + + (i) Oe root will e reiprol of the other if (ii) Oe root is zero if (iii) Roots re equl i mgitude ut opposite i sig if (iv) Both roots re zero if (v) Roots re positive if d re of the sme sig d is of the opposite sig (vi) Roots re of opposite sig if d re of opposite sig (vii) Roots re egtive if,, re of the sme sig () Let f ( + +, where > The (i) Coditios for oth the roots of f ( to e greter th give umer re ; f( ) ; >

5 Qudrti Equtios d Iequtios (ii) Coditios for oth the roots of f ( to e less th give umer re, f ( ) >, < (iii) The umer lies etwee the roots of f (, if > ; f( ) < (iv) Coditios for etly oe root of f ( to lie etwee d is f ( ) f( ) <, > (v) Coditios for oth the roots of f ( re ofied etwee d is f ( ) >, f( ) >, d < <, where < (vi) Coditios for oth the umers d lie etwee the roots of f ( is > ; f( ) < ; f( ) <

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