The complete algebra; embracing simple and quadratic equations, proportion, and the progressions, with an elemenary and practial view of logarithms, ... designed to be sufficiently eleme

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1877 Excerpt: ...terms of the root, we observe the following operations: (ti f» A. (o + 6)» =o» +3afb + 3o6« + b' =a3 +3a! + 3ao-f 66. B. (a + b + c)» = (a + 6) + c3 (a + 6)3 + 3(a + &) + 3(o + b)e + c«c = en m (s) a3 + 3a + 3ao + 6»6 + 3(a + &) + 3(a + b)c + c«c. C. (a + 6 + e + d)3 = (a + 6 + c) + d = (a + b + c)3 + 3(o + b + c)« + 3(a + 6 + c)/ + d')d = (1) (2) (S) a» + 3a' + 3o6 + &»6 + 3(a + 6) +3(a + b)c + cc + 3(a + & + c)» + 3(a + 6 + c)d + d»ti. Hence it appears; 1st, That the cube of a polynomial is made up of as many parts as there are terms in the root; 2nd, that the first part is the cube of the first term of the root; 3d, That the second part is three times the square of the first term of the root-f-3 times the first term info the second term--the square of the second term, multiplied by the second term of the root; 4th, That any one of the parts of the power, as the nth, is Three times the square of the n--1 preceding terms of the root,-f-3 times the product of these terms into the next, or nth term, + the square of this last or nth term, all these terms being multiplied by the last, or nth term of the root. Finally, it is evident that, if the work does not terminate by this process when the letter of arrangement disappears from the remainder, it can never terminate, since the divisor always contains this letter. Sch. 1.--If the first term of the arranged polynomial is not a perfect cube the root cannot be extracted. Sch. 2.--If at any time no term of the remainder is exactly divisible by the first term of the trial divisor, the root can not be extracted. Explanation--1st I arrange this polynomial with reference to o, and thus see at once the first two terms. But the terms 36a c ...