Elements of Algebra

254 ELEMENTS SECT. IV. we should immediately have its three roots. Let us, therefore, consider, in a more general manner, these three factors, $x-p$, $x-q$, $x-r$. Now, if we seek their product, the first, multiplied by the second, gives $x^2 - (p+q)x + pq$, and this product, multiplied by $x-r$, makes $x^3 - (p+q+r)x^2 + (pq+pr+qr)x - pqr$. Here, if this formula must become $=0$, it may happen in three cases: the first is that, in which $x-p=0$, or $x=p$; the second is, when $x-q=0$, or $x=q$; the third is, when $x-r=0$, or $x=r$. 722. Let us now represent the quantity found, by the equation $x^3 - ax^2 + bx - c = 0$. It is evident, in order that its three roots may be $x=p$, $x=q$, $x=r$, that we must have, 1. $a = p+q+r$, 2. $b = pq+pr+qr$, and 3. $c = pqr$. We perceive, from this, that the second term of the equation contains the sum of the three roots; that the third term contains the sum of the products of the roots taken two by two; and lastly, that the fourth term consists of the product of all the three roots multiplied together. From this last property we may deduce an important truth, which is, that an equation of the third degree can have no other rational roots than the divisors of the last term; for, since that term is the product of the three roots, it must be divisible by each of them: so that when we wish to find a root by trial, we immediately see what numbers we are to use *. For example, let us consider the equation, $x^3 = x+6$, or $x^3 - x - 6 = 0$. Now, as this equation can have no other rational roots than numbers which are factors of the last term 6, we have only 1, 2, 3, 6, to try with, and the result of these trials will be as follows: If $x=1$, we have $1-1-6 = -6$. If $x=2$, we have $8-2-6 = 0$. If $x=3$, we have $27-3-6 = 18$. If $x=6$, we have $216-6-6 = 204$. Hence we see, that $x=2$ is one of the roots of the given equation; and, knowing this, it is easy to find the other two; * We shall find in the sequel, that this is a general property of equations of any dimensions; and as this trial requires us to know all the divisors of the last term of the equation, we may for this purpose have recourse to the Table, Art. 66.