§ 3. SQUARE ROOTS OF MULTINOMIALS. 1. The square root of a trinomial which is the square of a binomial and the square roots of certain multinomials can be found by inspection (Ch. VIII., § 1, Art. 9). 2. Since we have (a + b)2 = a2 + 2 ab + b2, √(a2+2ab+b2) = a + b. From this identity we infer: (i.) The first term of the root is the square root of the first term of the trinomial; i.e., a = √a2. (ii.) If the square of the first term of the root be subtracted from the trinomial, the remainder will be Twice the first term of the root, 2 a, is called the Trial Divisor. (iii.) The second term of the root is obtained by dividing the first term of the remainder by the trial divisor; i.e., b = 2 ab 2 a (iv.) If twice the first term of the root plus the second term, 2a+b (the complete divisor), be multiplied by the second term, b, and the product be subtracted from the first remainder, the second remainder will be 0. The work may be arranged as follows: a2+2ab+b2 a + b Ex. Find the square root of 4x-12x2y+9 y2. The work, arranged as above, writing only the trial and the complete divisor, is: 3. When the multinomial is the square of a trinomial, the process of finding the root is an extension of the method of Art. 2. The multinomial whose root is required should be arranged to powers of a letter of arrangement. Since (a+b+c)2 = (a + b)2 + 2 (a + b) c + c2 we have √[(a2 + 2 ab + b2) + (2 a + 2 b + c)c] = a + b + c. The first two terms of the root are found by inspection, or by the method of Art. 2. The work may be arranged as follows: Observe that 2 ac is the first term of the remainder after subtracting (a+b)2= a2+2ab+b2. For, in finding the first two terms of the root we first subtracted a2 and then 2 ab + b2. Notice also that the complete divisor at any stage is twice the part of the root already found, plus the term last found. Ex. 1. Find the square root of 4x-12x+29 x2-30 x + 25. The work follows: 4x-12x+29 x2-30 x + 25 | 2x2-3x+5 20 x2-30 x + 25 4 x2-6x+5 Only the trial divisor and the complete divisor of each stage are written, the other steps being performed mentally. 4. The preceding method can be extended to find square roots which are multinomials of any number of terms. The work consists of repetitions of the following steps: After one or more terms of the root have been found, obtain each succeeding term, by dividing the first term of the remainder at that stage by twice the first term of the root. Find the next remainder by subtracting from the last remainder the expression (2 a+b) b, wherein a stands for the part of the root already found, and b for the term last found. EXERCISES III. Find the square root of each of the following expressions: 17. 26 6 ax5+ 15 a2x2 - 20 a3x3 + 15 a1x2 - 6 a3x + ao. 4 a 64 a6 - 64 a5 32 a3+48 a + 12 a2. 17 a2 - 22 a3 + 13 a4 - 24 a − 4 a5 + 16. 20. 9 x6 + 6 x3y + 43 x1y2 + 2 x3y3 + 45 x2y1 — 28 xyб + 4 yo. 22. a2mx2 + 10 a2m-2x2n+1 6 am+1xn+1 + 25 ɑ2m−4x2n+2 −30 am-1x+2 + 9 a2x2. § 4. CUBE ROOTS OF MULTINOMIALS. 1. The process of finding the cube root of a multinomial is the inverse of the process of cubing the multinomial. (i.) The first term of the root is the cube root of the first term of the multinomial; "i.e., a = Va3. (ii.) If the cube of the first term of the root be subtracted from the multinomial, the remainder will be 3a2b+3ab2 + b3, (3a2 + 3 ab + b2)b. = Three times the square of the first term of the root, 3 a2, is called the Trial Divisor. (iii.) The second term of the root is obtained by dividing the 3a2b first term of the remainder by the trial divisor; i.e., b = 3a2 (iv.) If the sum 3 a2 + 3 ab + b2, the complete divisor, be multiplied by the second term of the root, and this product be subtracted from the first remainder, the second remainder will be 0. (4) 3a2b+3ab2+b3 = (3a2+3 ab+b2) xb Ex. 1. Find the cube root of 27 x3 + 54 x2y + 36 xy2 + 8 y3. The work, arranged as above, is: 27 x2+54 x2y+36 xy2+8 y3 | 3x+2y (4) 54x2y+36 xy2+8 y3 | = (27 x2+18 xy +4 y2) (2 y) 2. The preceding method can be extended to find cube roots which are multinomials of any number of terms, as the method of finding square roots was extended. The work consists of repetitions of the following steps: After one or more terms of the root have been found, obtain each succeeding term by dividing the first term of the remainder at that stage by three times the square of the first term of the root. Find the next remainder by subtracting from the last remainder the expression (3 a2 + 3 ab + b2)b, wherein a stands for the part of the root already found, and b for the term last found. The given multinomial should be arranged to powers of a letter of arrangement. Ex. 27-27x+90 x2 − 55 x3 + 90 x1 − 27 x3 +27 x | 3−x+3x2 27 - 27 x 81 x2-54x3 3(3)2=27 3(3)2+3(3) ( − x) + ( − x)2 = 27 −9x+x2 3(3-x)2+3(3−x) (3 x2) + (3x2)2= 81x2-54x3+90 x4 − 27 x5 + 27 x 27-18 x + 30 x2 − 9 x3 +9x4 |