2. Find the cube of 2x-3y2. (2x-3y2)3 (2x)3 – 3(2x)2 (3y2)+3(2x) (3y2)2 — (3y2)3 = The results (1) and (2) are in accordance with the following law: The cube of a polynomial is equal to the sum of the cubes of its terms, plus three times the product of the square of each term by each of the other terms, plus six times the product of every three different terms. We will now prove by Induction that this law holds for the cube of any polynomial. Assume that the law holds for the cube of a polynomial of m terms, where m is any positive integer; that is, which, by (3) and Art. 230, is equal to (Art. 231, +3a+1(a+a2 + A3 + ··· + am) + ɑm+13 3 ... This result is in accordance with the above law. Hence if the law holds for the cube of a polynomial of m terms, where m is any positive integer, it also holds for the cube of a polynomial of m +1 terms. But we know that the law holds for the cube of a poly. nomial of three terms, and therefore it holds for the cube of a polynomial of four terms; and since it holds for the cube of a polynomial of four terms, it also holds for the rube of a polynomial of five terms; and so on. Hence the law holds for the cube of any polynomial. Example. Expand (223 — x2+2x−3)3. In accordance with the above law, we have (2x3-x2+2x-3)3 = = (2 x3) 3 + ( − ∞2) 3 +(2x)3+(=3)3+3(2x3)2(−x2+2x−3) +3(−x2)2(2x+2x−3)+3(2x)2 (2 x3 — x2 — 3) +6(2x)(-x) (2x) +6 (2 x3) ( − x2) (-3) +6(2x3) (2x) (-3)+6(-x2) (2x) (-3) =8x-x6+8x3-27-12x2+24x7-36x+6x2+6x3-9x4 +24x3-12x1— 36x2+54x3-27 x2+54x-24x6 +36x3-72x+36x3 =8x-12x+30x7-61x+662-93x+98 x3-63x2 +54x-27. EXAMPLES. 233. Expand the following; m and n being any positive 10. (-4x+6x2-4x+1). 16. (2x-3x-1)3. 13. (-2) 11. (2x2+5x)3. 17. (a-b-c+d)3. 12. (4a2x-3b3y)3. 8. (2x3-3x2- x — 4)3. XVI. EVOLUTION. 234. Evolution is the process of finding any root (Art. 121) of an expression. Note. We shall consider in the present chapter those cases only in which both the expression and its root are rational (Art. 154). 235. We may extend the definition of Art. 123, and say that any rational expression is a perfect power of the nth degree when it has a rational nth root. 236. Evolution of Monomials. We have already given (Art. 124) a rule for finding a root of a rational, integral, and positive monomial, which is a perfect power of the same degree as the index of the required root. We will now consider the general case. 1. Required the fourth root (Art. 122, Note) of 16x1y12. By Art. 109, either (2xy) or (-2xy) is equal to 16x1y12. Whence by Art. 121, either 2xy3 or -2xy is a fourth root of 16x1y12; a result which is expressed in the form √16 x1y12 = ±2xy3. Note. The sign ±, called the double sign, is prefixed to an expression when we wish to indicate that it is either + or -. 2. Required the fifth root of 243 a1015. Since (-3a2b3)5-243 a1b15, we have 5 243 a 115 — 3a2b3. == We then have the following rule for finding any root of a rational and integral monomial, which is a perfect power of the same degree as the index of the required root: Extract the required root of the absolute value of the numeri cal coefficient, and divide the exponent of each letter by the index of the root. Give to every even root of a positive term the sign ±, and to every odd root of any term the sign of the term itself. To obtain any root of a fraction each of whose terms is a perfect power of the same degree as the index of the root, extract the required root of both numerator and denominator, and divide the first result by the second. Thus, 3 27 asbe $27 a3be 3ab2 4c3 237. Square Root of a Polynomial. Let A and B be two rational expressions (Art. 154) arranged in the same order of powers (Art. 73) of some common letter, x; and let the exponent of x in the last term of A be greater than, or less than, its exponent in the first term of B, according as A and B are arranged in descending or ascending powers of x. By Art. 108, (A + B)2 = A2 + 2AB+ B2. (A + B)2 — A2 = 2 AB + B2. If the expression 2AB+ B2 is arranged in the same order of powers of x as A and B, its first term must be twice the product of the first term of A and the first term of B. Hence, the first term of B may be obtained by dividing the first term of the expression 2AB+ B2 by twice the first term of A. Note. The expression "first term of A," in the above discussion, is understood to mean the sum of all the terms of A containing the highest, or lowest, power of x, according as A is arranged in descending or ascending powers of x. Thus, if A= ax + bx + cx3, then the first term of A is (a+b)xa. A similar meaning is attached to the expressions "last term of A" and "first term of B." |