Elements of Algebra

112 ELEMENTS SECT. II. of this is evident; for since $-b$ is a term of the root, the powers of that letter will ascend in the following series, $-b$, $+b^2$, $-b^3$, $+b^4$, $-b^5$, $+b^6$, &c. which clearly shews that the even powers must be affected by the sign $+$, and the odd ones by the contrary sign $-$. 343. An important question occurs in this place; namely, how we may find, without being obliged to perform the same calculation, all the powers either of $a + b$, or $a - b$. We must remark, in the first place, that if we can assign all the powers of $a + b$, those of $a - b$ are also found; since we have only to change the signs of the even terms, that is to say, of the second, the fourth, the sixth, &c. The business then is to establish a rule, by which any power of $a + b$, however high, may be determined without the necessity of calculating all the preceding powers. 344. Now, if from the powers which we have already determined we take away the numbers that precede each term, which are called the coefficients, we observe in all the terms a singular order: first, we see the first term $a$ of the root raised to the power which is required; in the following terms, the powers of $a$ diminish continually by unity, and the powers of $b$ increase in the same proportion; so that the sum of the exponents of $a$ and of $b$ is always the same, and always equal to the exponent of the power required; and, lastly, we find the term $b$ by itself raised to the same power. If therefore the tenth power of $a + b$ were required, we are certain that the terms, without their coefficients, would succeed each other in the following order; $a^{10}$, $a^9b$, $a^8b^2$, $a^7b^3$, $a^6b^4$, $a^5b^5$, $a^4b^6$, $a^3b^7$, $a^2b^8$, $ab^9$, $b^{10}$. 345. It remains therefore to shew how we are to determine the coefficients, which belong to those terms, or the numbers by which they are to be multiplied. Now, with respect to the first term, its coefficient is always unity; and, as to the second, its coefficient is constantly the exponent of the power. With regard to the other terms, it is not so easy to observe any order in their coefficients; but, if we continue those coefficients, we shall not fail to discover the law by which they are formed; as will appear from the following Table.