RELATION BETWEEN THE ROOTS AND COEFFICIENTS OF EQUATIONS - UNIT - 1 - PART - 1- MATHS MAJOR

Añadido: 2026-05-20

[00:00:43]plus a x power a one x power one plus a naught in the expression polynomial okay so in the polynomial coefficients [Music] so in the polynomial function which is equal to this one is f of x which is equal to zero being another polynomial equation or algebraic equation okay so in the polynomial you [Music] root circle for example this equation consider this equation x square minus six x plus five equal to zero in the equation okay in minus five x square plus two x minus eleven equal to zero either within a maintenance cubic equation in the cubic equation [Music] square corner coefficient minus five hexagonal coefficient two constant okay so in the coefficient other one comma minus five comma two comma minus 11 in the coefficients [Music] so in the in the between roots and coefficients of equation right let the equation of the polynomial rather a naught x power n plus a 1 x power n minus 1 plus a 2 x power n minus 2 plus etcetera plus a n minus 1 x plus a n equal to [Music] some a 1 by a naught equal okay so some of the roots of the indra there minus a 1 divided by a naught of the a1 x power n minus 1 upon a coefficient summation alpha 1 equal to minus a 1 divided by a naught other sum of the roots of dinner summation alpha 1 means summation summation means sum of the roots so alpha 1 plus alpha 2 plus etc plus plus alpha n equal to minus a 1 divided by a naught minus a 1 divided by a naught okay [Music] [Music] some of the product of the roots i know [Music] [Music] that means alpha one alpha two plus alpha two alpha three plus etcetera plus alpha n minus one alpha n equal to a two by a naught abden okay okay so next point sum of the product of some of the product of the roots taken three at a time array time number value a3 minus a one by a naught sum of the product of taken [Music] on the foreign the last coefficient constant a n divided by a naught okay minus one power l so you don't know the general form one two one [Music] [Music] [Music] minus a 1 divided by a naught so alpha one or the value one plus alpha two or the value phi equal to minus a one order value minus six divided by a naught of the value one okay so five plus one six equal to minus into minus six either under equal average satisfaction so in the relation is true or not true you know sum of the roots so so next one render the condition sum of the product of roots taken which is equal to so sum of the product of roots taken to at a time and alpha 2 is 5 so 1 into 5 which implies a 2 a 2 what a value 5 divided by a naught or the value 1 okay 1 into 5 5 equal to 5 [Music] or in the arithmetic x cube right x cube x a comma a plus d eprin okay so given equation is p into x cube plus q into x square plus r into x plus five equal to zero a minus q cube plus 27 p square is equal to x cube minus 12 x square plus 39 x minus 28 equal to zero in the equation satisfying right so first condition first one some of the roots relation between roots and coefficients some of the roots so some of the roots have been rather equal a3 so a naught here a naught equal to p and a one equal to q and a two equal to r a three equal to five okay so in the sum of the roots of the inter there minus a one divided by a naught so some of the roots what are the roots here a a minus d a minus d plus a plus a plus d equal to minus a one a one out of one q so minus q divided by a naught a naught over the value p right so e power minus d plus t cancel idea a plus a plus c a 3 a 3 a equal to minus q divided by p this implies a equal to minus q divided by 3 p ok sum of the product of roots some of the product of product of roots say a two by a naught a two by a naught okay first one negative level and then second one positive term alternative one negative positive level so some of the product of roots and then second one a into a plus d third one a plus d into a minus d so a minus d into a plus a into a plus d plus a plus d into a minus g equal to a two by a naught that means a to what a value say a two by a two what a value in a square minus a d plus c a square plus a d plus a minus b a plus b for metallica so a square minus d square equal to r divided by p minus a d u plus a 18 cancel i don't know what the moon is square three a square minus d squared minus q divided by three p the whole square minus r divided by p equal to d square q square divided by 3 p square 1 minus r divided by p equal to d square this implies d square order value so unknown value of my left side which support say india three p l c m n r o three p so q square minus three p r divided by three p square so next one third point sum of the roots last one three root circle model so product of the roots so third one product of the roots product of the roots so product of the roots in the moon and root cinnamon product is is minus a square r sorry a into d square which is equal to 5 divided by p already a what a value d squared or the value number theory oh fine but neutral minus q the whole cube divided by 3 p the whole cube minus a into hey what a value minus q divided by 3 p into d square order value q square minus 3 p r divided by 3 p squared equal to five divided by p okay so simplify for n obtainer minus q cube divided by twenty seven p cube minus so denominator first multiplied cube q into q square q cube minus three p q r which is equal to phi divided by p okay so either on lcm numerator three multiplied per nucleon so whole divided by 27 p cube minus q q plus 3 q power 3 minus 9 p q r equal to five divided by p so problem foreign cube plus q into x square plus r into x plus yes okay graph so either quantum parallel so minus q power 3 plus 3 into q cube minus 9 p q or equal to s into 27 [Music] minus nine into p q r equal to 27 into h into p square okay so you put numerous equation in the equation so 27 yes into p square left side on the right side okay so minus so in the term minus in the same in the minus underside of quantum promoter plus item so 2 into q power 3 plus 27 p cube s equal to 9 p comparing so p into x cube plus q into x square plus r into x plus s equal to 0 with x cube minus 12 x square plus 39 x minus 28 equal to zero okay so in the equation so x cubed value another x cubed of coefficient p next cube of coefficient 1 so p equal to 1 and x square order coefficient q in the x squared order coefficient minus 12 x order coefficient r in the exodus coefficient 39 [Music] [Music] 2 this implies 2 into minus 12 the whole cube plus 27 into p cube p cube so 1 cube into s s order value minus 28 which is equal to 9 into p p order value 1 into q order value minus 12 r or the value 30 nine okay so either simplify panama there so minus a cube upon minus two into so q to 1 2 244 into 12 so 244 into 12 four twos are eight eight four four four so 8 2 so remaining 1 so 9 2 enter 2 minus 2 so total line gain of aroma so 16 remain one two twos are four five so nine twos are eighteen eight remaining one two twos are four five so minus five eight five six minus in the value 27 one cube one cubic on a value one q one so twenty eight into twenty seven so twenty eight into 27 multiplied banana so 27 into 28 8 7's are 15 6 so remaining 5 8 2s are 16 16 twenty one so fourteen remaining one so two twos are four five so six five seven so minus seven five six seven hundred and fifty six so either multiplied negative terminal answer one so nine enter nine into twelve so nine into nine into twelve so nine tools are eighteen one nine seven nine so one not eight into thirteen nine so eight nines are 72 remaining seven so 7 9 8 3's are 20 4 remaining 2 so 2 3 so 2 11 so 12 remaining one four so the multiplayer panel mother anglican answer minus three four five six three thousand four hundred and fifty six minus seven t six in the value which is equal to minus minus four to one two okay so either add funding minus four to one two which is equal to minus four to one two so satisfy [Music] and then product vano of dina minus q divided by 3 p minus q over value minus 12 divided by 3 into p order value 1 which implies minus into minus plus so 12 divided by three so four okay what a value for a equal to four so next defined so d square equal to q square minus 3 p r divided by 3 p square okay so which is equal to cubed value in an angle so minus 12 so minus 12 square minus 12 square minus 3 into pre-order value pre-order value 1 and then r order value 39 divided by 3 into p square 3 into p order value so d square this implies minus 12 minus 12 of the whole square minus square formula so 12 into 12 into 12 4 2 right so and then 1 2 2 1 1 4 4 1. so 144 minus 3 into 39 9 3's are 27 so seven remaining two three threes are nine it's eleven nine plus two eleven naruto minus divided by three into one square for now one another so divided by 3 so 144 117 sorry given the category 7 2 so 27 divided by 3 so 27 divided by 3 9 3's are 27 so d square order value 9 of the na d order value integra so this implies d equal to three d equal to square rooted plus or minus three okay so the value d equal to plus three minus d comma a comma a plus d this implies a value minus three so already or minus recurs so minus into minus plus 3 comma 4 comma 4 minus 3 upon 7 comma 4 comma 1 so 7 comma 4 comma 1 in the root sum a 1 comma 4 comma 7 so the cubic equation in the cubic equation in the equation in the equation one comma four comma seven so in the root s foreign salem contact nine zero four two zero three zero one six three future vision study center sks hospital road opposite to hotel prakash near yuba stand salem two sell three zero one six three