Elements of Algebra

CHAP. XI. OF ALGEBRA. 253 CHAP. XI. Of the Resolution of Complete Equations of the Third Degree. 719. An equation of the third degree is called complete, when, beside the cube of the unknown quantity, it contains that unknown quantity itself, and its square: so that the general formula for these equations, bringing all the terms to one side, is $ax^3 \pm bx^2 \pm cx \pm d = 0$. And the purpose of this chapter is to shew how we are to derive from such equations the values of $x$, which are also called the roots of the equation. We suppose, in the first place, that every such an equation has three roots; since it has been seen, in the last chapter, that this is true even with regard to pure equations of the same degree. 720. We shall first consider the equation $x^3 - 6x^2 + 11x - 6 = 0$; and, since an equation of the second degree may be considered as the product of two factors, we may also represent an equation of the third degree by the product of three factors, which are in the present instance, $(x - 1) \times (x - 2) \times (x - 3) = 0$; since, by actually multiplying them, we obtain the given equation; for $(x - 1) \times (x - 2)$ gives $x^2 - 3x + 2$, and multiplying this by $x - 3$, we obtain $x^3 - 6x^2 + 11x - 6$, which are the given quantities, and which must be $= 0$. Now, this happens, when the product $(x - 1) \times (x - 2) \times (x - 3) = 0$; and, as it is sufficient for this purpose, that one of the factors become $= 0$, three different cases may give this result, namely, when $x - 1 = 0$, or $x = 1$; secondly, when $x - 2 = 0$, or $x = 2$; and thirdly, when $x - 3 = 0$, or $x = 3$. We see immediately also, that if we substituted for $x$, any number whatever beside one of the above three, none of the three factors would become equal to 0; and, consequently, the product would no longer be 0: which proves that our equation can have no other root than these three. 721. If it were possible, in every other case, to assign the three factors of such an equation in the same manner,