All succeeding terms would involve 0 as a factor, and would there fore=0; :. (a+b)=a+6ab+15aab2+20a3b3 +15a2ba+6ab3 +bo. (3x — 2a)3 — (3x)3 — 5. (3x)5-1.2a+5.(5-1) (3x)5-2. (2a)2 + 5.(5-1).(5-2) 1.2 (3x)5-3. (2a)3 =243x-810x+a+1080x3a2-720x2a3 +240xα1-32a3. Ex. XXV. Examples for practice. 1. Expand the following expressions: (1) (a−b); (1+x)5; (2−3x)1; (a2—b2)3. (3) (a3—x3)1; (5a2bc-4ac2)1; (a3+x3)3; (a−1— y−1)3. 2. Find the coefficient of b3 in the expansion (a-b)7; and the coefficient of 12 in the expansion of (3a2 —7x3)8. 3. Write down the coefficient of x in the expansion of (1+2x--x2)7, and of 4 in the expansion of (1-ax+bx2)1. EVOLUTION. 50. Evolution is the inverse of Involution; being the method of finding any root of a given quantity. Evolution of Monomials. As the involution of any monomial is performed by multiplying the index of the given quantity by the whole number which indicates the proposed power, so the roots of a given monomial may be obtained by dividing the index of that quantity by the whole number which corresponds to the proposed root. In order therefore to extract any root of a given monomial, we must inquire what quantity multiplied into itself once, twice, or oftener, till the number of factors equals the number of units in the number which indicates the given root, will give the quantity whose root is to be extracted; or we may apply the following general Rule. RULE. "Extract the root of the numerical coefficients, if there be any, for the numerical part, and divide the indices of the letter or letters by the whole number which corresponds to the root required." therefore square root of 4x2y=2xy, or √(4x2y2)=2xy. Note 1. Any even root of a positive quantity may be positive or negative; thus the square root of a2 is either a or -a, since (+a) ×(+a) gives +a2, and (-a) × (-a) gives +a2; but an odd root of a positive quantity must be positive, since (+a) × (+ a) × (+ a) = +a3, but (-a)x(-a)x(-a) gives a3. Note 2. An even root cannot be assigned for a negative quantity; thus the square root of a2 cannot be extracted, because there is no real quantity, positive or negative, which will, when multiplied into itself, produce a2; and therefore even roots of negative quantities are called IMPOSSIBLE OF IMAGINARY quantities. They are expressed by means of the radical sign; thus the square root of a2 would be √(-a), or a √√(−1). Note 3. An odd root of any quantity will have the same sign as the quantity itself; thus, since -ax-a× -a - a3, therefore √/(-a3) = =-a. Note 4. If the quantity under the root does not admit of resolution into the number of identical factors denoted by the root, or in other words, if it be not a complete power, then its exact root cannot be extracted, and the quantity itself, with the sign annexed, is called a SURD (Art. 19): thus √35, √a2, 1/363, are surd quantities; they are also called IRRATIONAL quantities, whereas quantities, whose roots can be exactly determined, are called RATIONAL quantities. Evolution of Binomials, Trinomials, &c. 51. In order to extract the roots of compound quantities, we must observe how the terms of the roots may be obtained from those of the power. Since the expression (a+b) (a+b) = a2 +2ab+b2 is true for all values of a and b; we may determine a general rule for finding the square root by observing the manner in which a+b may be obtained from a2+2ab+b2. Arrange the terms according to powers of a. a2+2ab+b2 (a+b a2 2a+b 2ab+b2 2ab+b2 We observe that the first term of a2+2ab+b2 is the square of a, the first term of the root. Put therefore a for the first term of the root, square it, and then subtract the square from the first term of the power. There now remains 2ab+b2, to find b. Bring down twice the first term of the root; i. e. 2a for a trial divisor; divide 2ab by it, and the quotient is b, the other term of the root; which place in the root after a, with a positive sign affixed; add also b to the trial divisor, i.e. make the divisor 2a+b, and let this divisor be multiplied by b, the second term of the root; the product is 2ab+b2; put this under the two terms brought down above, and on subtraction nothing remains. If more terms remain, consider a+b as one quantity, and also notice that its square a2+2ab+b2 has already been subtracted from the given quantity; proceed to find the next term of the root by the process mentioned above; i.e. bring down 2 (a+b) for a trial divisor, divide the remainder by twice this new value of a for the next term of the root; if it be c, then add c to the terms already found of the root and to the trial divisor; then multiply 2a+2b+c by c, as in the former process, and put down the product under the terms brought down above, and subtract. This process may be continued till the entire root is found, or an approximation to it. Note. In examples in evolution the terms of the expression must be first arranged according to the descending powers of some letter involved. Ex. XXVII. A. Ex. 1. Extract the square root of a2 — 4ab+4b3. Proceeding according to the Rule given above. |