If we make r + r'+ r'' = n', and p''' + r' + &c. = s'', we shall have N=(n'+s'')2 = n'2 + 2n's" + s''2; and N— n'2 = R" = 2 (r + po' + po'') (p''' + p2 + &c.) + 8" 2; in which, if we perform the operations indicated, the term 2r" will contain a higher power of the leading letter than any following term. Hence, If we divide the first term of the third remainder by twice the first term of the root, the quotient will be the fourth term of the root. If we continue the operation, we shall see, generally, that The first term of any remainder, divided by twice the first term of the root, will give a new term of the required root. It should be observed, that instead of subtracting n2 from the given polynomial, in order to find the second remainder, that that remainder could be found by subtracting (2r+r')" from the first remainder. So, the third remainder may be found by subtracting (2n+r")" from the second, and similarly for the remainders which follow. Hence, for the extraction of the square root of a polynomial, we have the following RULE. I. Arrange the polynomial with reference to one of its letters, and then extract the square root of the first term, which will give the first term of the root. Subtract the square of this term from the given polynomial. II. Divide the first term of the remainder by twice the first term of the root, and the quotient will be the second term of the root. III. From the first remainder subtract the product of twice the first term of the root plus the second term, by the second term. IV. Divide the first term of the second remainder by twice the hrst term of the root, and the quotient will be the third term of the root. V. From the second remainder subtract the product of twice the sum of the first and second terms of the root, plus the third term, by the third term, and the result will be the third remainder, from which the fourth term of the root may be found as before. VI. Continue the operation till a remainder is found equal to , or till the first term of some remainder is not divisible by twice the first term of the root. In the former case the root found is exact, and the polynomial is a perfect square; in the latter case, it is an imperfect square. EXAMPLES. 1. Extract the square root of the polynomial 49a2b2-24ab3 + 25 aa · 30a3b + 16b1. First arrange it with reference to the letter a. 25a+b2 — 40a3b2c + 76a2b2c2 — 48ab2c3 + 36bc4 30abc + 24a3bc2 Remarks on the Extraction of the Square Root of Polynomials. 1st. A binomial can never be a perfect square. For, its root cannot be a monomial, since the square of a monomial will be a monomial l; nor can its root be a polynomial, since the square of the simplest polynomial, viz., a binomial, will con tain at least three terms. Thus, an expression of the form a2 + b2 can never be a perfect square. If so, 2d. A trinomial, however, may be a perfect square. when arranged, its two extreme terms must be squares, and the middle term double the product of the square roots of the other two. Therefore, to obtain the square root of a trinomial, when it is a perfect square, Extract the square roots of the two extreme terms, and give these roots the same or contrary signs, according as the middle term is positive or negative. To verify it, see if the double product of the two roots is equal to the middle term of the trinomial. also, But 2 x 3a3 X(-8ab2)=-48a4b2, the middle term. 4a2+14ab +962 is not a perfect square: for, although 4a2 and +962 are perfect squares, having for roots 2a and 36, yet 2 × 2a x 36 is not equal to 14ab. Of Radical Quantities of the Second Degree. 102. A radical quantity is the indicated root of an imperfect power of the degree indicated. Radical quantities are sometimes called irrational quantities, sometimes surds, but more commonly, simply radicals. The indicated root of a perfect power of the degree indi cated, is a rational quantity expressed under a radical form. An indicated square root of an imperfect square, is called a radical of the second degree. An indicated cube root of an imperfect cube, is called a radical of the third degree. Generally, an indicated nth root of an imperfect nth power, is called a radical of the nth degree. Thus, 2, 3 and 6, are radicals of the second degree; 3/4, 3/18 and 3/11, are radicals of the third degree; and /4, 1/5 and 1/11, are radicals of the nth degree. The degree of a radical is denoted by the index of the root. - The index of the root is also called the index of the radical. 103. Since like signs in both factors give a plus sign in the product, the square of a, as well as that of +a, will be a2: hence, the square root of a2 is either a or a. Also, the square root of 25a2b4 is either + 5ab2 or -5ab2. Whence we may conclude, that if a monomial is positive, its square root may be affected either with the sign + or ; √9a1 = ±3a2, thus, for,+3a2 or 3a2, squared, gives 9a1. The double sign ±, with which the root is affected, is read plus or minus. If the proposed monomial were negative, it would have no square root, since it has just been shown that the square of every quantity, whether positive or negative, is essentially positive. Therefore, such expressions as, are algebraic symbols which indicate operations that cannot be performed. They are called imaginary quantities, or rather, imaginary expressions, and are frequently met with in the solution of equations of the second degree. Generally, Every indicated even root of a negative quantity is an imaginary expression. An odd root of a negative quantity may often be extracted For example, -27=-3, since (- 3)3 — — 27. 3 = Radicals are similar when they are of the same degree and the quantity under the radical sign is the same in both. Thus, ab and c, are similar radicals of the second degree. Of the Simplification of Radicals of the Second Degree.\ 104. Radicals of the second degree may often be simplified, and otherwise transformed, by the aid of the following principles. 1st. Let the a, and, denote any two radicals of the second degree, and denote their product by p; whence, Squaring both members of equation (1), (axiom 5), we have, (√a)2 × (√ ̃)2 = p2, or, ab = p2 (2). Extracting the square root of both members of equation (2), (axiom 6), we have, √ab=p; but things which are equal to the same thing are equal to each other, whence, The product of the square roots of two quantities is equal to the square root of the product of those quantities. 2d. Denote the quotient of a by√, by q; whence, Squaring both members of equation (1), we find, Extracting the square root of both members of equation (2), we have, = q. |