EVOLUTION OP A POLYNOMIAL] 



MATHEMATICS. ALGEBRA. 



481 



CHAPTER V. 

 ALGEBRA. MISCELLANEOUS INVESTIGATIONS. 



A FEW particulars connected with the elements of 

 algebra remain to be noticed. These might have been 

 introduced earlier, and are so introduced in most books 

 on the subject. We have thought it expedient, how- 

 ever, with a view to facilitate the progress of the stu- 

 dent, to postpone them to this place. 



To Extract the Square Root of a Polynomial 



When we set about extracting the square root of a 

 quantity, whether a number or an algebraical expression, 

 we proceed on the supposition that the proposed quan- 

 tity is really a square ; that is, that it is capable of being 

 produced from two equal factors. You know, that in 

 numbers, this is not always the case ; 2, 7, 8, 10, <kc., 

 are not squares, and therefore, in strictness, have no 

 square roots ; by the aid of decimals, however, we can 

 approximate, as it is called, to the square root of any of 

 these numbers ; that is, we can find a number such that 

 'the square of it shall differ from the number proposed 

 by as small a decimal as we please ; so, that for all 

 practical purposes, the root, thus determined, may be 

 used for that of the proposed number ; since it is the 

 exact root of a number differing from the proposed one, 

 only by a decimal too minute to bo appreciated. The 

 difference here spoken of would be actually exhibited by 

 the remainder left at the close of the operation ; for, by 

 subtracting this remainder from the number proposed, 

 wo should convert that number into a complete square ; 

 we always take care, in such arithmetical operations, to 

 push the approximative process sufficiently far to render 

 the correctional remainder of no moment, in reference 

 to the practical inquiry in hand. 



It is the same with algebraical expressions or 

 polynomials ; we proceed (by the rule presently to be 

 given) on the supposition that the square root actually 

 exists, carrying on the process, as in arithmetic, till the 

 terms of the polynomial have all been used up. If a 

 remainder be left, we conclude that the polynomial Is 

 not a square, and can assign the correction necessary to 

 make it a square ; for this correction, as in arithmetic, 

 is the remainder. In algebra, it is usual to stop as soon 

 as the proposed polynomial is exhausted ; the object 

 being, in general, more to ascertain whether the ex- 

 4on submitted to the process is a square or not, 

 than to seek algebraical approximations to imperfect 

 squares. Indeed, the term algebraical approxirnatioa 

 would bo meaningless, as you must see ; for, in the 

 absence of all numerical interpretation of our symbols, 

 how could we speak of a set of symbolical expressions 

 (i/'/n-oximtitiny, as to value, to any other such expres- 

 sion ? When numerical values are given to the letters, 

 we may with propriety speak of such approximations ; 

 but, even then, what are approximations for some nume- 

 rical values, will be departures for others. 



Let us now take a complete algebraic square, and try 

 to discover by what process its root may be evolved. 

 And first, let the square be that of a binomial; namely, 

 the square of a + 6, which is a 2 -f- 2 ab -f- 6 2 . 



Now write down this expression as in the margin, and 

 mark off a place for the 



root, to the right. The * + 2oi + 6 2 (a + 6 



square root of the first 

 term is a, which we know 

 to be the first term of the 

 sought root ; and subtract- 



ing o 2 from the given expression, we get 2nb + 1- for a 

 remainder. Consider this remainder as a diridend, and 

 let us see whether, by help of the term o, already in the 

 root, we cannot discover the leading term of a divisor for 

 this dividend, suited to give 6, the other term of the 

 root. A glance at the dividend shows that the proper 



VOL. I. 



2a + 6] 



2ab + 



leading term is just double the partial root a, already 

 obtained ; we therefore write 2a for the first term of the 

 divisor we are seeking to form ; and as, in algebra, the 

 first term of a divisor is all we want, to get a term of the 

 quotient, we at once pronounce 6 to be the second term 

 of the quotient or root ; and, to complete the divisor, 

 we put this 6, not only in the quotient, but also in the 

 divisor ; and we find that 6 times the complete divisor, 

 2a + 6, equals the dividend; and thus the operation 

 terminates, and the required root is discovered. 



Guided by this easy process, let us now endeavour to 

 evolve the root from the square of a trinomial ; namely, 

 from the square of a + 6 + c, which is (a + 6) 3 + 

 2(a+ b)c + c 3 ; this being evidently the square of 

 (a + 6) + c. Writing the polynomial in the usual way, 

 it would be a 2 + 2ab + b- + 2(a + 6) c + c 2 ; and the 

 following method, for discovering the terms a, 6, c, of 

 the root, one after another, is in exact imitation of the 

 operation above. 



a' 4. 2 ab + i' + 2 (a + *) c + c' [a + 6 + c 



2 ab -\- b' 

 2 ab -{- 4* 



2 (a + 4) c -4- c' 

 2 (a + i) c + c 7 



Here the first term a of the root is found as before : 

 wo then get the remainder or dividend 2ab + b- ; the 

 double of the known portion a of the root furnishes the 

 trial or incomplete divisor 2a, by help of which the 

 second term 6, of the root is found ; after which tho 

 divisor is completed by joining this 6 to the incomplete 

 form : we then get a second remainder or dividend 

 2(a + b) c + c-, and take the double of what is now in 

 tho root for the corresponding incomplete divisor : this 

 gives for quotient c the third term of the root ; and 

 the addition of this c completes the divisor, the multi- 

 plication of which by the quotient-term c, finishes the 

 process, as no remainder is left. 



In like manner, by applying the same proceeding to 

 the square of a quadrmomial a -J- 6 + c + d, we should 

 obtain the several terms of the root, one after another ; 

 the polynomial whose root is sought being written in 

 the form 



a 5 + (2 a + t) b -f (2(a + b) + c) c + d"; 



as a mere contemplation of this form, in connection 

 with the above process, makes sufficiently evident. And 

 as the square of any polynomial (a + 6+c + d + . +<)"> 

 may be written 



the process is general : it is expressed in words as 

 follows : 



RULE 1. Arrange the terms of the proposed poly- 

 nomial as if for division, marking off a place as for 

 quotient. 



2. Find the square root of the leading term, put it 

 in the quotient's place, and the square of it under tho 

 leading term, which will of course be equal to it. Draw 

 a line, as if for subtraction, and bring down, under it, 

 the next two terms of the polynomial : these will form 

 a dividend, and a place, to the left of it, is now to be 

 marked off, for a divisor. 



3. Put twice the root- term, just found, in tho divisor's 

 place : see how often this incomplete, divisor is contained 

 iu tho leading term of the dividend, and connect the 

 quotient, with its proper sign, both to the root-term, 

 and to the incomplete divisor : the divisor will thus bo 

 completed. 



3 Q 



