When the index of the power to which the quantity is to be raised, is a large number, this process of continued multiplication would evidently become extremely tedious and inconvenient. We shall therefore explain the method of finding the powers of compound algebraic quantities by Sir ISAAC NEWTON'S Binomial Theorem. This theorem admits of a general demonstration, but at present it may be sufficient to deduce it from an induction of a few particular facts. In the involution of a binomial quantity of the form a+b, the powers of each of the letters, together with their coefficients, are found to observe certain laws, which will be readily understood by an examination of the following table: F 5th Power. (a+b)5 a+5ab10a3 b2 + 10 a2 b3 + 5 a b1 + b5 6th Power. (a+b)6 a66a5b+15 aa b2 + 20a3 b3 +'15 a2 b2 + 6 a b3 + bo The successive powers of ab, are precisely the same as those of a+b, except that the signs of the terms will be+ alternately. and 49. In reviewing the preceding table, where several of That in any one of those powers, the first term a of the the powers of a+b are expanded, it may be observed, 1st, binomial appears in every term except the last, and the last term b of the binomial in every term except the first. 2d, That the index of a, in the first term of the power, is the same with the index of the power to which the binomial is to be raised, and that the indices of the other powers of a, in the successive terms, continually diminish by unity to the last term but one, where it is unity itself. 3d, That the index of b, in the second term of the power, is unity, and that the indices of b, in the successive terms, continually increase by unity to the last term, where it is the same with the index of the power to which the binomial is to be raised. From this statement, it will not be difficult, leaving the coefficients out of the question, to express the literal parts of any power of a+b; suppose, for example, the 7th power. The powers of a and b separately, will be as follows. Or, combining these into the form of multiplication, a", ab, ab2, a*b3, a*b*, a2b3, ab", b Next, as to the coefficients of these terms, 1st, It may be observed, on examining the table, that the first term which has a coefficient, is the second term of the power; and that this coefficient is the same as the index of the first term, or of the power to which the binomial is to be raised. Thus, in the square it is 2; in the cube 3; and in the fourth power 4, &c. 2d, That if the coefficient of a in any term be multiplied by its index in that term, and the product divided by the number of terms to that place, reckoning from the left, the quotient will give the coefficient of the next term. Thus, in the 6th power, 6×5 30 2 = 2 = Coeff. of a in the 2d term x its index 15 coefficient of the third term. In the same Coeff. of a in the 3d term x its index manner, Number of terms to that place = 15 X 4 60 3 = 3 20 coefficient of the fourth term, and so on for the other coefficients. By applying this same rule throughout, the coefficient of the last term will be unity, which is not expressed. Hence we are furnished with general rules for raising a binomial to any power required, without the laborious process of actual multiplication. Let it be required to find the 7th power of a+b. The terms without the coefficients, as already found, are a2 ab a3b2 a1b3 a3b± a2b3 ab6 b7 Hence (a+b)=a7+7a©b+21a5 b2+35a*b3+35a3b1+21a2b5+ 7ab+b. In the same manner, it will be found, that (a—b)”— a7a6b+21 a5 b2 — 35 a1b3 +35a3b*— 21 a2b5+7ab6 b7. b2. (a+b)10a10+10a3b+45a8b2+120a1 b3+210a6 b*+252a5 b5+ 210a4b6+120a3 b2+45a2b3+10ab9+b1o. — It may here be remarked, that when the index of the power is an odd number, the number of terms (being always greater by one than the units in its index) will be an even number and conversely. In the former case, the two coefficients in the middle will be the same, and those coefficients which are equally distant from the middle ones, will be equal. In the latter case, when the index of the power is an odd number, the greatest coefficient will be in the middle, and will have equal coefficients at equal distances from it. By attending to this law of the coefficients, they need only be computed as far as the middle term or terms, and then the same written in an inverted order. Thus, the coefficients of (a+b)" are 7, 21, 35, 35, 21, 7. (a+b)10 are 10, 45, 120, 210, 252, 210, 120, 4.5, 10. 50. By attending to the rules which have been stated, the binomial theorem may be presented in a more general form. Suppose it were required to raise the binomial (a+b) to the power whose index is n. Proceeding with n as we have done with the indices in the preceding examples, it appears, that The use of this general form of the theorem, will appear in its application to an example or two. Ex. 1. Required the 4th power of 3x2+2y. In this instance, n = 4; a= 3x2, and b= 2y, and by substituting these in the general formula, we have, Hence n× (n−1)(n—2) 2X3 2 4X3X2 2X3 ×3x2 × (2y)3=96x2y3. is (2y)1 = 16y+. (3x2+2y)1=81x3+216x6y+216x1y2+96x2y3 +16y*. Ex. 2. Required the 7th power of 2x-7. 72+ Hence, n=7; a=2x, and 67; and by the formula, (2x-7)=(2x) + 7 × (2x)° x -7 +21 x (2x)5 × 35x (2x)+x-73 +35 × (2x)3 x-7+21 x (2)2 x 75+ 7x (2x) X-76-77-128x7-3136x+32928x192080x+ 672280x3-1411788x2+1647086x-823543. |