EVOLUTION. 179
Rearrange in descending powers of x. 16x4 -24x% + 25x2a2_ i2xa-3 + 4a4(4x2 - 3x«+ 2a2 16x4
- 2 - 3 xa
xa + 2 a2 24 x% + 25x2^2 24x%f 9x2^2 16x-V- 12x«3 + 4a4 16 x2rt2 _ 12 a-«3 + 4 «4
Explanation. When we have obtained two terms in the root, 4x2 — 3 xa, we have a remainder 16x2a2_ i2xa3 + 4«4.
Double the terms of the root already found and place the result, 8x2— 6x«, as the first part of the divisor. Divide 16x2^2^ the first term of the remainder, by 8x2, the first term of the divisor; we get + 2 «2 which we annex both to the root and divisor. Now multiply the complete divisor by 2a2 and subtract. There is no remainder and the root is found.
EXAMPLES XXI. b.
Find the square root of each of the following expressions:
1. x2-10x?/ + 25?/2. 6. 4x4-12x3+29x2-30x + 25. 2. 4x2-12x?/+ 9?/2. 7. 9x4 -12x3- 2x2 4-4X + 1. 3. 81 x2 + 18 xi/ + ^2. 8. x4 -4x3 + 6x2-4x+ 1. 4. 25x2-30xy + 9?/2. 9. 4«4 + 4^3 _ 7^,2 _ 4^ _f. 4. 5. a4 - 2 a3 + 3 (^2 _ 2 a + 1. 10. 1 - 10 x + 27 x2 - 10 x^ + x*. 11. 4x2 + 9^/2 + 25^2 4- I2x?/ -30?/^-20x^. 12. 16 x6 + 16 x^ - 4 x8 - 4x9 + x^. 13. x6 - 22 x4 + 34 x3 + 121 x2 - 374 x + 289. 14. 25 x4 - 30 ax3 + 49 dP-x'^ - 24 ciH + 16 a*. 15. 4 X* + 4 x2?/2 - 12 X202 + ?/4 - 6 ?/202 + 9 z 16. 6 obH - 4 a^hc + arl)^ + 4 «2c2 + 9 /^2c2 _ 12 a&c2. 17. - 6 62c2 + 9 c4 + ?)4 _ 12 c2a2 + 4 a4 + 4 a?-b'^. 18. 4 X4 + 9 ?/4 + 13 x22/2 _ 6 X^3 _ 4 a:3y. 19. 1 -4x+ 10x2 -20x3 + 25x4-24x5 + 16x6. 20. 6 acx^ + 4 62^:4 + «2a;io + 9 c2 _ 12 6cx2 - 4 a&x^.
199. When the expression whose root is required contains fractional terms; we may proceed as before, the fractional
