INVOLUTION OF BINOMIALS. 128. A BINOMIAL can be raised to any power by successive multiplications. But when a high power is required, the operation is long and tedious. The BINOMIAL THEOREM, first developed by Sir Isaac Newton, enables us to expand a binomial to any power by a short and speedy process. 129. In order to investigate the law which governs the expansion of a binomial we will expand a + b and a -b to the fifth power by multiplication. By examining the different powers of a+b and a-b in these Examples, we shall find the following invariable laws governing the expansion : 1st. The leading quantity (i. e. the first quantity of the binomial) begins in the first term of the power with an exponent equal to the index of the power, and its exponent decreases regularly by one in each successive term till it disappears; the following quantity (i. e. the second quantity of the binomial) begins in the second term of the power with the exponent one, and its exponent increases regularly by one till in the last term it becomes the same as the index of the power. Thus, in the fifth power the Exponents of a are 5, 4, 3, 2, 1. Exponents of bare 1, 2, 3, 4, 5. It will be noticed that the sum of the exponents of the letters in any term is equal to the index of the power. 2d. The coefficient of the first term is one; of the second, the same as the index of the power; and universally, the coefficient of any term, multiplied by the exponent of the leading quantity, and this product, divided by the exponent of the following quantity increased by one, will give the coefficient of the succeeding term. Thus, in the fifth power, 5, the coefficient of the second term, multiplied by 4, a's exponent, and divided by 5 X 4 = 10, the coeffi 1 plus 1, b's exponent plus 1, = cient of the third term. 2 The coefficients are repeated in the inverse order after passing the middle term or terms, so that more than half of the coefficients can be written without calculation. The number of terms is always one more than the index of the power; i. e. the second power has three terms; the third power, four terms; and so on. When the number of terms is even, i. e. when the index of the power is odd, the two central terms have the same coefficient. 3d. When both terms of the binomial are positive, all the terms of the power are positive; but when the second term is negative, those terms which contain odd powers of the following quantity are negative, and all the others positive; or every alternate term, beginning with the second, is negative, and the others positive. Having found the preceding coefficients and the coefficient of th middle term, we can write the others at once. Hence, x8 +8x7 y +28 x6 y2 + 56 x5 y3 + 70 x1 y1 + 56x3 y5 + 28 x2 y6 +8x yì + y3. 2. Expand (a - b). Ans. ao—6a3b+15a3b2—20a3b3+15a2ba—6ab5+bo. 6. Expand (b+c)1. 7. Expand (x + 1)5. NOTE. Since all the powers of 1 are 1, 1 is not written when it appears as a factor; but its exponent must be used in obtaining the coefficients. Ans. x55x + 10 x3 + 10 x2 + 5 x + 1. 8. Expand (1— y)o. Ans. 1-6y+15 y2-20 y3 + 15 y — 6y+yo. 9. Expand (a —— 1)3. 130. When the terms of the binomial have coefficients or exponents other than 1, the theorem can be made to apply by treating each term as a single literal quantity. In the expansion, each factor should be enclosed in a parenthesis, and after the expansion of the binomial by the binomial theorem, the work should be completed by the expansion of the enclosed factors, according to the rule for the expansion of monomials. 1. Expand (2 x — y2)1. OPERATION. (2x)1 — 4 (2 x)3 (y2) + 6 (2 x)2 (y2)2 — 4 (2 x) (y2)3 + (y2)1 Expanding each factor as indicated, we have 16 x 32 x3 y2+24x2 y* — 8 x y + y3 - 2. Expand (3x2-2y)5. (3x2)5—5 (3x2)4 (2 y) + 10 (3x2)3 (2y)2 — 10 (3x2)2 (2 y)3 + 5 (3x2) (2y) — (2 y)5. Ans. 24310-810xy+1080x°y-720 x y + 240xy-32y. NOTE. Any letters, as a and b, might be substituted for 3x2 and 2y, and the expansion of (a - b) written out, and then the values of a and b substituted. 3. Expand (a2 — 3 b)1. Ans. a 12 a b + 54 a1 b2 108 a2 68+81 64. 4. Expand (x2 — y2)5. |