Page:Elementary algebra (1896).djvu/196

178 ALGEBRA. down as the first term of the required root. Subtract a2 from the given expression and the remainder is 2ab + b2 or (2a + b) b.

Thus, b, the second term of the root, will be the quotient when the remainder is divided by 2a + b.

This divisor consists of two terms: 1. The double of a, the term of the root already found.

2. b, the new term itself. The work may be arranged as follows:

o^ 2a--b 2 ab + b^ 2 ab + ?>2

Ex. 1. Find the square root of 9x^ — 42xy -- i9y

9x^-i2xy + 49 y^{Sx-7y 9^ 6x-7 y -42xy + 49 y^ -42xy + 49 y^

Explanation. The square root of 9x2 is 3x, and this is the first term of the root.

By doubling this we obtain 6x, which is the first term of the divisor. Divide -42xy, the first term of the remainder, by 6x and we get -7y, the new term in the root, which has to be annexed both to the root and divisor. Next multiply the complete divisor by -7y and subtract the result from the first remainder. There is now no remainder and the root has been found.

The process can be extended so as to find the square root of any multinomial. The first two terms of the root will be obtained as before. When we have brought down the second remainder, the first part of the new divisor is obtained by doubling the terms of the root already found. We then divide the first term of the remainder by the first term of the new divisor, and set down the result as the next term in the root and in the divisor. We next multiply the complete divisor by the last term of the root and subtract the product from the last remainder. If there is now no remainder the root has been found; if there is a remainder we continue the process.

Ex. 2. Find the square root of 25 x2a2 _ 12 xa^ + IG x* + 4 a^ - 24:X^a.