Elements of Algebra

$x^3 + 36x^2 + 432x + 1728$; the product also of the sum of these cubes by the difference $12$, is $12(2x^3 + 36x^2 + 432x + 1728) = 102144$; and, dividing successively by $12$ and by $2$, we have $x^3 + 18x^2 + 216x + 864 = 4256$, or $x^3 + 18x^2 + 216x = 3392 = 8 \cdot 8 \cdot 53$. If now we substitute $x = 2y$, and divide by $8$, we shall have $y^3 + 9y^2 + 54y = 8 \cdot 53 = 424$. Now, the divisors of $424$ are $1, 2, 4, 8, 53, \&c.$ $1$ and $2$ are evidently too small; but if we make $y = 4$, we find $64 + 144 + 216 = 424$. So that $y = 4$, and $x = 8$; whence we conclude that the two numbers sought are $8$, and $(8 + 12) = 20$. 732. Question 5. Several persons form a partnership, and establish a certain capital, to which each contributes ten times as many pounds as there are persons in company: they gain $6$ plus the number of partners per cent; and the whole profit is $392$ pounds. Required how many partners there are? Let $x$ be the number required; then each partner will have furnished $10x$ pounds, and conjointly $10x^2$ pounds; and since they gain $x + 6$ per cent, they will have gained with the whole capital, $\frac{x^3+6x^2}{10}$, which is to be made equal to $392$. We have, therefore, $x^3 + 6x^2 = 3920$; consequently, making $x = 2y$, and dividing by $8$, we have $y^3 + 3y^2 = 490$. Now, the divisors of $490$ are $1, 2, 5, 7, 10, \&c.$ the first three of which are too small; but if we suppose $y = 7$, we have $343 + 147 = 490$; so that $y = 7$, and $x = 14$. There are therefore fourteen partners, and each of them put $140$ pounds into the common stock. 733. Question 6. A company of merchants have a common stock of $8240$ pounds; and each contributes to it forty times as many pounds as there are partners; with which they gain as much per cent as there are partners. Now, on dividing the profit, it is found, after each has received ten times as many pounds as there are persons in the company, that there still remains $224l$. Required the number of merchants? If $x$ be made to represent the number, each will have contributed $40x$ to the stock; consequently, all together will have contributed $40x^2$, which makes the stock