

MATHEMATICS. ALGEBRA. [BXTRACTIOX or inn SQUARB ROOT. 



4. Multiply tho complete divisor by the root-term 



Sit fount), subtract the product from tho dividend, nnd 

 the remainder unite the two next term* of the poly- 

 nomial ; and a ttcond dividend will be obtained. 



Proceed with thi* as with the former, marking off a 

 place for a new divisor, and putting in that place twice 



the root-quantity already found, for on ineompUU divuor, 

 by aid of which tho third root- term may bo found, which, 

 added to tho incomplete divisor, renders it com, 

 And this uniform process is to bo continued till all tho 

 terms of the polynomial havo been brought down, aa in 

 tht fallowing examples : 



+ 1C* 8* + 4 (3*' 2* + 2 



6*-2*] 12x+16x 

 I2*>+ 4*' 



C* tr-i-2] 



12* 8* + 4 

 12* 8* + 4 



2. 4** 24*' + CO* 4 80* + CO* 1 24* + 4(2*> f * + 6* 2 

 4* 



00* 



4* 1 



4a J 12** + 12* 2] 



24* 80* + 60*' 

 24* 72JJ + 36* 1 



3. a- 

 a'- 



+ + 6a' + 4a- 

 _ 2a- + "] 40 s " + + 6a + 



' 24* + 4 

 8*> + 24* 24* + 4. 



+ 3 + a 4 (a' OT 2a l * + n 



2o* 4a+" 



-f 



4. 4* 4* 3* + 2* + 4 (2* * 1 

 4*< 



4*' *] 4* 3* 



4*' 2* 1] 4*' + 2x + 4 

 - 4*' + 2* + 1 



In this example there is a remainder, after all tho 

 terms of the proposed polynomial have been used : we 

 infer therefore that tho polynomial is not a complete 

 square ; but that it would be made one, by subtracting 

 3 (the remainder) from it : the expression 2x 2 z I 

 is the complete square root of 4x* 4x 3 3x 2 + 2x + 1 : 

 the polynomial proposed cannot be produced from two 

 Kfual factors : it is the product of tho unequal factors 

 (3x x 1+ V 3)(2x x 1 V 3). 



EXAMPLES FOR EXERCISE. 



Extract the square root of each of the following ex- 

 premions : 



1. * + 4 4* + 4 4. Root, x + 2 b. 



2. 9*'+12*'+10* 1 + 4f + 1. Riot, 3*'+2* + l. 



3. 9**+12* > +34* 1 +20* + 25. Root, 3 *' + 2 * + 5. 



4. * + 4 * + 2 * 4 + 9 r* 4 *+4. Root, *"+2 * *+2. 



5. ** 4 **+10 x*4 * 7 *'+24 *+16. Root, *" 2 *" 



+ 3*+4. 



6. 



Roof, 2*'+ f *+5. 



In this last example you may, if yon please, multiply 

 the polynomial by 4, in order to get rid of tho fraction : 

 the square root of the result will, of conno, be the 

 square root of tho given expression, multiplied by tho 

 square root of 4 : that is, by 2 ; you must, therefore, 

 remember to divide tho root by 2. 



The above general rule, for the square root of an 

 algebraical expression, applies, of cotirso, to number* ; 

 and augment* the arithmetical operation given at page 449, 

 in tho Arithmetic. Tho first step in this operation, is 

 to separate the figures of the number into periods, by 



marking off two figures, commencing at tho units' placo, 

 then two to tho left of those, and HO on. This enable? 

 UH to determine the local value of the leading figure of 

 tho root, and thence the number of intoger-places in the 

 >lote root. Thus, taking for example tho number 

 ;ti96, we see, by help of tho periods, that the 

 first figure of the root must bo in the place of (Aoiuam/x, 

 the square of it being so many millions, liko tho 7<> in 

 tho lirst period. The detailed operation, as suggested 

 by tho Algebra, is therefore as follows : 



76,80 76 9G(8000+700+60-H= 

 Cl 00 00 00 8700+60+4= 



16000+700=10700)12 80 76 96 

 1 1 69 00 00 



8760+4=8764 



17400+60=17460) 1 11 7696 

 1 01 76 00 



17520 + 4 = 17524) 



70096 

 7 00 96 



Tho above operation is the same, in principle, as that 

 at page 449 of tho Arithmetic ; by a reference to which 

 you will see that the moro compact form, which the 

 work assumes in the place r<>iYrri-<l to, arises merely from 

 tho suppression of the useless ciphers or ?i 1 the 



postponement of tho successive periods till they are 

 actually wanted in the several dividends. 



To Extract the Square Root of a Binomial, one of whose 



term* is Rational, and the other a Quadra 

 It sometimes happens that expressions of tho form 



