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The Conditions for Multiple Roots in Cubic and Quartic Equations

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The Conditions for Multiple Roots in Cubic and Quartic Equations Fort Hays State University FHSU Scholars Repository Master's Theses Graduate School Summer 1953 The Conditions For Multiple Roots In Cubic and Quartic Equations Laurence Dryden Fort Hays Kansas State College Follow this and additional works at: https://scholars.fhsu.edu/theses Part of the Algebraic Geometry Commons Recommended Citation Dryden, Laurence, "The Conditions For Multiple Roots In Cubic and Quartic Equations" (1953). Master's Theses. 508. https://scholars.fhsu.edu/theses/508 This Thesis is brought to you for free and open access by the Graduate School at FHSU Scholars Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of FHSU Scholars Repository. THE CONDITIONS FOR MULTIPLE ROOTS IN CUBI C AND QUARTIC EQUATIONS being A Thesis Presented to the Graduate F aculty of the Fort Hays Kansas State College in Partial Fulfillment of the Requireme nts for t h e De gree Master of Science by Laurence A. Dryden, B.S. in Education Ohio State University Approved~,R;I;_#~ Major Departme Date 7- '-?.. - ..,-.3 ACKNOWLEDGlvIENT The writer of this thesis wishes to acknowledge all sources of information used in its preparation and is especially indebted to the Head of the Mathematics Department, Professor Emmet C. Stopher, for his assistance, criticisms, and advice. \ TABLE OF CONTENTS CHAPTER PAGE INTRODUCTION 1 I. DISCRil'HNANT 1. Definition of Discriminant ••• . 2 2. Definition of Resultant 2 3. Relation between Discriminant and Resultant 3 4. Definition of Sylvester's Determinant D(f,g) •••• 3 S. Relation between Sylvester's Determinant and Resultant 6. Relation between Discriminant and Sylvester's Determinant 7 7. Proof of Quadratic ••.•• 8 8. Discriminant of Cubic Equation 10 9. Discriminant of Quartic Equation. 12 10. Meaning of Sign of Discriminant 19 II. MULTIPLE ROOTS OF CUBIC EQUATIONS 1. Condition for Two Equal Roots 24 2. Condition for Three Equal Roots 24 A. Geometric Interpretation •• 24 B. Algebraic Conditions for Three Equal Roots . 30 CHAPTER PAGE III. MULTIPLE ROOTS OF QUARTIC EQUATIONS 1. Conditions for Two Equal Roots. 39 2. Conditions for Three Equal Roots • • • 39 A. Geometric Interpretation. 39 B. Algebraic Conditions • • 4S 3. Conditions for Four Equal Roots SO A. Geometric Interpretation. B. Algebraic Development IV. SUMNlARY •• . BIBLIOGRAPHY • INTRODUCTION The solving of equations of the third and fourth order is often a very complex problem. It is helpful to know as much about the character of the roots as possible before a solution is attempted. Therefore, t h e problem of this thesis is to derive a set of equations involving the coefficients of the third and fourth order equations that can be used as a test to determine if the equation has any multiple roots. Equations that have real coefficients are the most common type. The problem would become much more complicated if equations which have complex coefficients were included. Therefore, the problem of this thesis is limited to the study of conditions for multiple roots of cubic and quartic equations having real coefficients. CHAPTER I DISCRIMINANT 1. Definition of Discriminant Let f(x)=axm+bxm-1 ••• be a polynomial of degree m whose factored form is: The discriminant~ off is defined to be a2m-2 times the product of the squares of the differences of r 1 , ..• , rm. The chosen power of a is the lowest power of a which eliminates fractions when~ is expressed as a polynomial in the coefficients of f(x). 2. Definition of Resultant Consider two polynomial equati ns (a 0 ~0, b 0 ~0) such that f(x)=O is lmown to have m roots Equations (2) have a root in common if and only if So as not to have fractions when this product is evaluated as a function of a 0 , ••• , bn, it is necessary to multiply this product by The resultant off and g is defined to be: 3 , ••• , equivalent polynomial in a 0 bn. 3. Relation between Discriminant and Resultant By differentiating (1), and substituting r 1 , r 2 , ••• , rm in succession for x, it is found that: 1 f (r1 )=a(r1-r2 )(r1-r3 ) ••• (r1 -rm) f'(r )=a(r -r )(r -r ) ••• (r -rm) ..................................2 2 1 2 3 2 1 In the second line replace r 2 -r1 by -(r1-r2 ). In f (r3) 2 replace (r3-r1 )(r3-r2 ) by (-1) (r1-r3)(r2-r3 ). In f'(rm) replace each rm-ri by -(ri-rm) . Mult iplication now g ives: am-lf'(rl) ••• f'(rm)=(-1)1+2+ ••• +m-la~. The sum in the exponent of (-1) is equal to m(m-1)/2. By (3) the left member is the resul~ant of f(x) and f'(x). Thus: (4) ~=(-l)m(m-l)/2 .lR(f,f'). a 4. Definition of Sylvester's Determinant D(f,g). 1 Consider the equations: f(x)=a xm+a xm-l+ +a =0 0 1 m and g(x)=boxn+b1xn-l+ 1 Nelson B. Conkwri ght , Introduction to the Theory of Equations (Boston: Ginn and Company, 1941T, p. 162. 4 A determinant has as many rows as columns. To have as many rows as columns the equations f=O, xf=O, x2f=O, ••• , xn-lf=O and g=O, xg=O, x 2g=O, ••• , xm-lg=O are used. The coefficients of these equations can then be used to make the following determinants ao al a2 am 0 ............ 0 0 ao al am-1 am 0 ...... 0 0 0 a a a a o•• 0 0 m-2 m-1 m ......................................... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • am bo bl b2 bn 0 ............ 0 0 bo bl bn-1 bn 0 ...... 0 0 0 b 0 ••• bn-2 bn-1 bn o•• 0 . • • • • ••••••••••••••••••••••• ••••••••••••• bn In this determinant the coefficients of f(x)=O appear inn rows and the coefficients of g(x)=O appear in m rows, making the order of the determinant to be m+n. The above method of setting up a determinant was first developed by Sylvester and is known as Sylvester's determinant, D(f,g). 5 5. Relation between Sylvester's Determinant and Resultant Consider the equations: f(x)=a xm+a xm-l+a xm-2+ 0 1 2 Let the value of Sylvester's determinant be D. Consider the equation: ao al a2 am 0 ...................... 0 0 ao al . am-1 am 0 ................. 0 0 0 ao am-2 am-1 am 0 ............ 0 . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • am O 0 0 · · · · · · · · · · · · • · · · • • · · · · • · · • • • · · · · • • · • am-1 · · · · • · · · · · · · · · · · • • · · · • · · · · · • · · · · · · · · am-2 am-1 0 bo bl b2 b n -z 0 ...................... 0 bo bl bn-1 bn-z 0 ................. 0 0 0 bo bn-2 bn-1 bn-z 0 ............ 0 . .. • bn - z 0 0 . bn-l bn-z 0 .................................... b n-2 b n-1 b n -z 6 Laplace's development2 of this determinant by rows gives: pzm+qzm-1+ ••• +D=O where p=(-l)ma~. The values of the coefficients of the zm-1 term and subsequent terms are not needed in the proof and the constant term Dis the value of the determinant for z=O. Denote t h e roots of f(x)=O by r 1 , r 2 , ••• , rm and let r be any root. Let z=g(r). To t h e elements of the last column add r m+n- l times the elements in the first column plus rm+n-2 times the elements in t h e second column plus rm+n-3 times the elements in the third column, etc. This does not change the value of the determinant . The elements of the last column starting at t h e top are: +amrn-l=rn-l[f(r))=O a rm+n-2+a rm+n-3+ +a rn-2=rn-2 [f(r))=0 0 1 m nth row element is: a rm+a rm-l+ ••• +am=f(r)=O 0 1 (n+l)th row element is: borm+n-l+blrm+n-2+ •·· +bnrm-l_zrm-l=rm-l[g(r)-z) nd (n+2) row element is: b rro+n-2+b rm+n-3+ ••• +b rm-2_zrm-2=rm-2[g(r)-z] o 1 n 2 Ibid., p. 124. 7 ........................................ (m+n) th row element is: b orn+b 1rn-1 + ·•• The first n elements are zero for r is any root of f(x). The last m elements will be zero if z=g(r1 ) or g(rm) . Therefore the determinant is zero for g(rm) , or the roots of equation (-l)magzm+ But the product of the roots of an mth degree equation always equals (-l)m times the constant term divided by the coefficient of the first term, from which D(f,g)=a~g(r1 )g(r2 ) •.. g(rm)=R(f,g) or ( 5) D(f,g)=R(f, g). 6. Relation between Discriminant and Sylvester's Determinant3 The purpose of this development has been to find a method of finding 6 by using Sylvester's determinant. Equation (4) g ives 6 in terms of the resultant off and f'. Equation (5) shows the relation between the resultant and Sylvester's determinant. 3 Ibid., pp. 166-7. 8 (4) 6=(-l)m(m-l)/2 ·1R(f a , f') Since g(x)=O is any equation, f'(x)=O could be used and equation (5) then becomes, D(f,f')=R(f,f 1 ). Substituting D(f,f') for R(f,f') in equ ation (4) gives the relationship, (6) 6=(-l)m(m-l)/2 .ln(f,f'). a The discriminant can be computed much easier by this method than by the formula usually g iven in Theory of Equations textbooks. 7. Proof for Quadratic Consider the general quadratic f(x)=ax2+bx+c=0 In nearly all alg ebra texts the exp ression b 2-4ac, which appears under the radical sign in the quadratic formula, is defined to be the discriminant of the quadratic equation.
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