Note on the General Quartic

M. E. Graber, A.M., Heidelberg University, Tiffin, Ohio.

American Mathematical Monthly, Vol. XVI (1909), 27-29.

The general quartic equation

(1)  ϕ(x) ≡ a_{0}x^{4} + a_{1}x^{3} + a_{2}x^{2} + a_{3}x + a_{4} = 0,

where the coefficients are all real, can be reduced to the form

(2)  ϕ(x) ≡ ax^{4} + bx^{2} + c = 0,

by the following method.

Suppose the quartic resolved into the factors:

(a_{1}x^{2} + b_{1}x + c_{1})  and  (a_{2}x^{2} + b_{2}x + c_{2}).

Then effect the rational bilinear transformation x = \frac{αy+β}{y+1} , obtaining

ϕ(x) = [\frac{a_{1}(αy + β)^{2}}{(y + 1)^{2}} + \frac{b_{1}(αy + β)}{y + 1} + c_{1}] [\frac{a_{2}(αy + β)^{2}}{(y + 1)^{2}} + \frac{b_{2}(αy + β)}{y + 1} + c_{2}] = 0.

In this expression α and β may be chosen so that the coefficients of the first powers of y shall be zero, after clearing of fractions.

The values of α and β fulfilling this condition are easily found to depend upon

(3)  αβ = \frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}} ,

(4)  α + β = 2\frac{a_{2}c_{1} − a_{1}c_{2}}{a_{1}b_{2} − a_{2}b_{1}} .

That α and β are real, is determined from the equation

(5)  t^{2} + 2\frac{(a_{2}c_{1} − a_{1}c_{2})}{(a_{1}b_{2} − a_{2}b_{1})}t + \frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}} = 0,

by means of the discriminant relation

(6)  \begin{aligned}△ & ≡ [\frac{2(a_{2}c_{1} − a_{1}c_{2})}{(a_{1}b_{2} − a_{2}b_{1})}]^{2} − 4 [\frac{b_{1}c_{2} − b_{2}c_{1}}{a_{1}b_{2} − a_{2}b_{1}} ] \\ & = \frac{4 [(a_{2}c_{1} − a_{1}c_{2})^{2} − (b_{1}c_{2} − b_{2}c_{1})(a_{1}b_{2} − a_{2}b_{1})]}{(a_{1}b_{2} − a_{2}b_{1})^{2}}. \end{aligned}

If v_{1}, w_{1}, and v_{2}, w_{2} are the two pairs of roots of the quartic, we may assume, without loss of generality, that v_{1} is greater than either v_{2} or w_{2} and w_{1} is greater than either v_{2} or w_{2}. From the two quadratic factors whose roots are, respectively, v_{1}, w_{1}, and v_{2}, w_{2}, we obtain

(7) −\frac{b_{1}}{a_{1}} = v_{1} + w_{1}, \frac{c_{1}}{a_{1}} = v_{1}w_{1}, −\frac{b_{2}}{a_{2}} = v_{2} + w_{2}, and \frac{c_{2}}{a_{2}} = v_{2}w_{2},

and these values substituted in the discriminant give for the determinations of its signs,

(8)  \begin{matrix}a_{1}^{2}a_{2}^{2}(v_{1} − v_{2})(v_{1} − w_{2})(w_{1} − v_{2})(w_{1} − w_{2}) \\ = (a_{2}c_{1} − a_{1}c_{2})^{2} − (b_{1}c_{2} − b_{2}c_{1})(a_{1}b_{2} − a_{2}b_{1}). \end{matrix}

If all the roots are real it is evident from (8) that the discriminant is positive. Likewise, if two roots are real and two imaginary, and also if all four roots are imaginary, the discriminant is seen to be positive, the conjugate pairs being, v_{1}, w_{1}, and v_{2}, w_{2}. Hence, (5) has real roots, and consequently α and β are real.

In the exceptional case where \frac{b_{1}}{a_{1}} = \frac{b_{2}}{a_{2}} , the substitution

x = y − \frac{b_{1}}{2a_{1}} = y − \frac{b_{2}}{2a_{2}} ,

gives directly ϕ(x) = A_{1}y^{4} +A_{2}y^{2} +A_{3}, where A_{1}, A_{2}, A_{3} are real constants. Finally, from (3), (4) and (7),

(9)  αβ = \frac{(v_{2} + w_{2})v_{1}w_{1} − v_{2}w_{2}(v_{1} + w_{1})}{(v_{1} + w_{1}) − (v_{2} + w_{2})},

(10)  and α + β = \frac{2v_{1}w_{1} − 2v_{2}w_{2}}{(v_{1} + w_{1}) − (v_{2} + w_{2})}.

From these equations it is easily seen that

2αβ − (α + β)(v_{1} + w_{1}) = −2v_{1}w_{1},

and 2αβ − (α + β)(v_{2} + w_{2}) = −2v_{2}w_{2}.

But these are the characteristic equations for harmonic pairs. Therefore, the pair α, β is harmonic with respect to the pairs v_{1}, w_{1} and v_{2}, w_{2}.