Elements of Algebra

CHAP. XI. OF ALGEBRA. 387 CHAP. XI. Of the Resolution of the Formula $ax^2 + bxy + cy^2$ into its Factors. 162. The letters $x$ and $y$ shall, in the present formula, represent only integer numbers; for it has been sufficiently seen, from what has been already said, that, even when we were confined to fractional results, the question may always be reduced to integer numbers. For example, if the number sought, $x$, be a fraction, by making $x = \frac{t}{u}$, we may always assign $t$ and $u$ in integer numbers; and as this fraction may be reduced to its lowest terms, we shall consider the numbers $t$ and $u$ as having no common divisor. Let us suppose, therefore, in the present formula, that $x$ and $y$ are only integer numbers, and endeavour to determine what values must be given to these letters, in order that the formula may have two or more factors. This preliminary inquiry is very necessary, before we can shew how to transform this formula into a square, a cube, or any higher power. 163. There are three cases to be considered here. The first, when the formula is really decomposed into two rational factors; which happens, as we have already seen, when $b^2 - 4ac$ becomes a square. The second case is that in which those two factors are equal; and in which, consequently, the formula is a square. The third case is, when the formula has only irrational factors, whether they be simply irrational, or at the same time imaginary. They will be simply irrational, when $b^2 - 4ac$ is a positive number without being a square; and they will be imaginary, if $b^2 - 4ac$ be negative. 164. If, in order to begin with the first case, we suppose that the formula is resolvible into two rational factors, we may give it this form, $(fx+gy) \times (hx+ky)$, which already contains two factors. If we then wish it to contain, in a general manner, a greater number of factors, we have only to make $fx+gy = pq$, and $hx+ky = rs$; our formula will then become equal to the product $pqrs$; and will thus necessarily contain four factors, and we may increase this number at pleasure. Now, from these two equations we obtain a double value for $x$, namely, $x = \frac{pq-gy}{f}$, and $x = \frac{rs-ky}{h}$. cc2