ANALYTIC SOLUTION OF QUARTIC AND CUBIC POLYNOMIALS By A J Helou, BCE, M.Sc., Ph.D. August 995
CONTENTS REAL AND IMAGINARY ROOTS OF CUBIC AND QUARTIC POLYNOMIALS. INTRODUCTION. COMPUTER PROGRAMS. REAL AND IMAGINARY ROOTS OF CUBIC POLYNOMIALS.. Rel roots.. Imginry roots.4 REAL AND IMAGINARY ROOTS OF QUARTIC POLYNOMIALS.4. Rel roots.4. Imginry roots REFERENCES APPENDIX A FINDING THE REAL ROOTS OF CUBIC POLYNOMIALS IN EUCLID S GEOMETRY
REAL AND IMAGINARY ROOTS OF CUBIC AND QUARTIC POLYNOMIALS. INTRODUCTION The nlytic solution presented in this pper, my e used to find the rel nd imginry roots of cuic nd qurtic polynomils in the form of + + + = (cuic) () 4 + + + + = (qurtic) () Where,,, nd re the rel coefficients of the cuic nd qurtic polynomils. The leding coefficients re tken s. If they re not, they should e mde y dividing the entire eqution y tht coefficient.. COMPUTER PROGRAMS The nlytic solution for the determintion of the rel nd imginry roots of qurtic nd cuic polynomils nd the computer progrms sed thereon were developed y the uthor nd pulished y Hewlett-Pckrd, N.E. Circle Blvd., Corvllis, Oregon 97, USA. The relevnt HP Users Lirries nd the pulished mteril re detiled s follows ) Rel roots only HP-67/97/4 Users Lirry. Solution ws registered under ctegory No. L45, Ctlogue No. 785C4, April 9, 984. The progrm ws developed for HP 4C clcultors. HP-75 Users Lirry. Solution ws registered under ctegory No. L45, Ctlogue No. 7554, My 7, 984. The progrm ws developed for HP-75C computers. ) Rel nd imginry roots HP-75 Users Lirry. Solution ws registered under ctegory No. L45, ctlogue # 75 54, Mrch, 986. The progrm ws developed for HP-75C computers.. REAL AND IMAGINARY ROOTS OF CUBIC POLYNOMIALS.. Rrel roots A generl form of cuic polynomils my e written s
y + y + y + = () where,, nd re rel coefficients Let y, where is vrile.then, (4) 7 let nd then eqution (4) ecomes 7 + + = (5) ) Trigonometric solution Constructed in Fig., re circle nd n ngle t its centre equl ß. Angle ß is divided into three equl ngles, ech equlß. Cords opposite to nglesßnd to ß re constructed forming similr tringles. From the formed similr tringles, the following reltions my e written BC OB CD BC nd BC OD CD FD hence = rz nd z r z c where, c = AB. By sustitution, we get - r + r c = (6) Eqution (5)=eqution (6) if nd only if = - r nd = r c Therefore, r nd c (7) From trigonometry, the following reltions my e derived hence, Therefore, r sin nd c r sin (8) sin
4 sin (9) The first rel root of eqution (5) is sin sin () The other two rel roots nd my then e determined s follows 4 & () This solution my e pplied if nd only if < nd () Whence, the roots of eqution (), y, y nd y my e determined s follows y, y nd y () ) Algeric solution If the conditions in eqution () re not stisfied then the cuic polynomil hs one rel root. The method for finding it is essentilly tht given y Hudde in 65. By trnsformtion, we get z, where z is vrile. z Then, y sustitution, eqution (5) ecomes 6 z z (4) 7 hence, z nd z 4 7 4 7 By sustitution for, we get z z 4 7 4 7 (5)
5 whence, the rel root y of eqution (), my now e determined s y (6) c) Geometric solution Appendi A offers solution in Euclid s geometry, for finding the rel roots of cuic polynomils tht stisfy the conditions of eqution ()... Imginry roots If rel root is found y eqution (5), the cuic polynomil hs then two more imginry roots tht my e determined s follows Divide eqution () y y - y y y y y y y y y y y y y y y y Since y is rel root then nd, y y y y y y y y The imginry roots y i nd y i re then determined s follows y y y 4 y y & y i i j (7) where j. Note tht cuic polynomil might hve the following numer of roots Three rel roots, or One rel nd two imginry roots
6.4 REAL AND IMAGINARY ROOTS OF QUARTIC POLYNOMIALS.4. Rel roots A generl form of qurtic polynomil is written s follows () 4 + + + + = (8) Where,,, nd re the rel coefficients. The leding coefficient is tken s. If it is not, it should e mde y dividing the entire eqution y tht coefficient. Eqution (8) my e fctored into two qudrtic polynomils s follows A C B D A C B D (9) Where, A, B, C nd D re so fr unknown coefficients for the qudrtic epressions. Multiplying the qudrtic epressions y ech other leds to 4 A A C B AB CD B D () Eqution ()=eqution (8) if nd only if = A () = A + B - C () = (AB - CD) () = B - D (4) Solving simultneously equtions () to (4) leds to A (5) B B B 4 8 8 (6) C B (7) 4 D B (8) Eqution (6) is cuic eqution in its generl form. To solve it let B y, where y is vrile. This leds to the following eqution 6
7 y Let y (9) 4 4 8 8 8 q 4 nd then, n lgeric solution for B gives r 4 8 8 8 r r q q r r q B y 6 4 7 4 7 6 () If the quntity under the squre root is negtive then trigonometric solution for B gives B q cos () 6 where, 7r cos q The determintion of B leds to the determintion of C nd D in equtions (7) nd (8). Thus the previously unknown coefficients A, B, C nd D ecome now known nd re epressed in terms of,, nd. The rel roots, my therefore e determined s follows The st nd nd roots nd, re A C A C 4B D & () The rd nd 4th roots nd 4, re A C A C 4B D & () 4.4. Imginry roots If the quntities under the squre roots in equtions (7) nd (8) re negtive, then the coefficients C nd D ecome imginry. No solution is offered for qurtic polynomils with imginry coefficients. Coefficients A, B, C nd D must ll e rel. If the quntities under the squre roots in equtions () nd () re negtive, then the qurtic polynomil hs imginry roots tht my e determined s follows
8 The st nd nd imginry roots i nd i, re A C A C 4 B D & i i j (4) The rd nd 4th imginry roots i nd i4, re A C A C 4 B D & i i 4 j (5) where j. Note tht qurtic polynomil might hve the following numer of roots Four rel roots, or Two rel nd two imginry roots, or Four imginry roots REFERENCES HP - 4C Mth Pc, Hewlett-Pckrd, N.E. Circle Blvd., Corvllis, Oregon 97, USA, 979.
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