Page:Elementary algebra (1896).djvu/201

EVOLUTION. 183 Ex. 2. Find the cube root of 27 + 108x + 90x2 - 80x3 - 60x4 + 48x5 - 8x6,

274108%+ 9022— 80a3— 60a!+ 482°— 8at(8 +42 —222 27 3x)? =27 108a+ 9022— 8023 8x38x4x= 4362 (40)2= +1622 27 + 8624 1692 | 1084+ 144924 6423 8x (8+ 4x)2 = 274+ 72% + 4822 — 549?— 14443— 6041+ 48 25— 846 3x(8+42) x(—222)= — 18 42 — 2493 (— 2%)2 = + 4a 27 + 7244 302? — 2493 4 4a4 54x2 - 144x3 - 60x4 + 48x5 - 8x6

Explanation. When we have obtained two terms in the root, 3 + 4x, we have a remainder

- 54x2 - 144x3 - 60x4 + 48x5 - 8x6,

Take 3 times the square of the root already found and place the result, 27 + 72x + 48x2, as the first part of the new divisor. Divide -54x2, the first term of the remainder, by 27, the first term of the divisor; this gives a new term of the root -2x2 To complete the divisor we take 3 times the product of (3 + 4x) and -2x2, and also the square of -2x2. Now multiply the complete divisor by -2x2 and subtract; there is no remainder, and the root is found.