Further Conepts for Avne Mthemtis - FP1 Unit Ientities n Roots of Equtions Cui, Qurti n Quinti Equtions Cui Equtions The three roots of the ui eqution x + x + x + 0 re lle α, β n γ (lph, et n gmm). The solutions to the eqution re x α, x β n x γ. To fin the reltionships etween the oeffiients in the originl eqution n the roots, we hve to use ifferent tehnique. Sine the solutions to the eqution re x α, x β n x γ, the eqution must hve the ftors ( x α), ( x β ) n ( x γ ). Multiplying these together woul give the first term x rther thn x. It follows tht the tul ftoristion of x + x + x + is ( x α)( x β )( x γ ) This gives us the ientity x + x + x + ( x α)( x β )( x γ ) multiplying out the right-hn sie gives x + x + x + x + x + x + x + x + x + ( x α )( x βx γx + βγ ) ( x βx γx + βγx αx + αβx + αγx αβγ ) x ( α + β + γ ) x + ( αβ + αγ + βγ ) x αβγ Equting oeffiients of x gives ( α + β + γ ) Hene α + β + γ the sum of the roots Equting oeffiients of x gives ( αβ + αγ + βγ ) Hene αβ + αγ + βγ the sum of the pirs of roots Equting the onstnt terms gives αβγ Hene αβγ the prout of the roots You shoul notie tht there re some similrities etween these formule for ui equtions n those for qurti equtions. There is formul for the sum n prout 1
just like the qurti equtions ut there is lso formul for the sum of ll the possile ifferent pirings of the roots. You will fin tht new formul ppers every time we inrese the orer of the eqution we look t. For qurti eqution, there will e four formule; the sum of the roots, the sum of the pirs of the roots, the sum of the triples of the roots n the prout of the roots. Nottion To ope with the onfusion tht oul rise, the following nottion is use α is use to stn for the sum of the roots For ui eqution, For qurti eqution, α α + β + γ α α + β + γ + δ αβ is use to stn for the sum of the pirs of roots For ui eqution, For qurti eqution, αβ αβ + αγ + βγ αβ αβ + αγ + αδ + βγ + βδ + γδ αβγ is use to stn for the sum of the triples of roots For qurti eqution, αβ γ αβγ + αβδ + αγδ + βγδ Exmples with the roots of ui equtions 1. The roots of the ui eqution x + x + x + 0 re α, β n γ. Fin the ui equtions with roots i) α, β n γ ii) 1 1 1, n α β γ For the originl eqution α the sum of the roots is αβ the sum of the pirs of the roots is αβγ the prout of the roots is
i) The sum of the new roots is α + β + γ α + β + γ 6 This is ( ) 1 α 6 6 The sum of the pirs of new roots is ( α )( β ) + ( α )( γ ) + ( β )( γ ) whih simplifies to αβ α β + + αγ α γ + + βγ β γ + αβ + αγ + βγ ( α + β + γ ) + 1 ( αβ ) ( α ) + 1 + 6 + 1 0 + 1 The prout of the new roots is ( α )( β )( γ ) whih simplifies to ( α )( βγ β γ + ) αβγ αβ αγ + α βγ + β + γ 8 αβγ ( αβ ) + ( α ) + 8 6 8 1 So the new eqution is 8 1 1 x x + 0x 0 1 1 x + x + 0x + 0 x + 1x + 0x + 1 0 ii) The sum of the new roots is 1 1 1 βγ + αγ + αβ αβ + + α β γ αβγ αβγ The sum of the pirs of new roots is 1 1 1 1 1 1 1 1 1 + + + + α β α γ β γ αβ αγ βγ
Using ommon enomintor, this simplifies to 1 1 1 + + αβ αγ βγ γ + α + β αβγ α αβγ ( ) ( ) The prout of the new roots is 1 1 1 1 α β γ αβγ The new eqution is x x + x 0 x + x + x + 0 x + x + x + 0 Compre this to the originl eqution x + x + x + 0 Do you notie nything? Di this hppen for qurti equtions (look k t the exmples on ) Cn you mke generlistion out mking new eqution using the reiprols of the originl roots? Qurti n Quinti Equtions Rules for the roots of qurti n quinti equtions n e foun y following the sme ies s for qurti n ui equtions. For the qurti eqution x + x + x + x + e 0, the roots re lle α, β, γ n δ (lph, et, gmm n elt). The solutions to the eqution re x α, x β, x γ n x δ. For ny qurti eqution: α αβ the sum of the pirs of roots i.e. αβ + αγ + αδ + βγ + βδ + γδ αβγ this is the sum of the omintions of roots i.e. αβγ + αβδ + αγδ + βγδ
αβγδ e For the quinti eqution x + x + x + x + ex + f 0, the roots re lle α, β, γ, δ nε (lph, et, gmm, elt n epsilon). The solutions to the eqution re x α, x β, x γ, x δ n x ε. For ny qurti eqution: α αβ the sum of the pirs of roots i.e. αβ + αγ + αδ + αε + βγ + βδ + βε + γδ + γε + δε αβγ this sum of the triples (work these out for yourself!) e αβγδ the sum of the omintions of roots f αβγδε This n mke working with equtions of higher orer iffiult ut there is quik metho for fining some of the new equtions tht re se on the roots of n originl eqution. The Sustitution Metho This metho n e use to fin new equtions where the sme thing is eing one to eh of the originl roots. E.G. The roots of z + z z + 1 0 re α, β n γ. Fin the ui eqution with roots α, β n γ If the originl eqution is written in terms of z, we n write new eqution in terms of nother vrile w, where w z. If w z then w z. We n sustitute this into the originl eqution giving w w w + + 1 0 7 7w + 9w w 9 6w + w w + + 1 0 8 7 7w + 9w w 7 18w + w w 1 + + + 1 0 7 7w + 9w w + 7 18w + w + 10w 0 + 0
w + 1w w + 8 0 w 1w + w 8 0 6