New Developments of Algebra

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Grant, Faires & Rodgers, 1876
 

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Σελίδα 87 - Corollary 4. — Every equation of an odd degree has at least one real root, and if there be but one, that root must necessarily have a contrary sign to that of the last term.
Σελίδα 87 - Every equation of an even degree whose last term is negative has, at least, two real roots; and if there be but two, one of these will be positive and the other negative.
Σελίδα 51 - Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.
Σελίδα 51 - RULE. 1. Separate the given number into periods of three figures each, beginning at the units place.
Σελίδα 87 - Corollary 4. — Hence, if the signs of the alternate terms be changed, and if p, and every quantity greater than p, renders the result positive, then — p is less than the least root.
Σελίδα 23 - To find two numbers such, that if the square of each be added to their product, the sums shall be both squares. Ans. 9 and 16 17.
Σελίδα 66 - The first derived polynomial will contain the factor n(x—a)"~' • that is. x—a occurs (n—1) times as a factor in the first derived polynomial. The greatest common divisor of the given equation and its first derived polynomial must therefore contain the factor (x—a) repeated once less than in the given equation. To determine, therefore, whether an equation has equal roots, find the greatest common divisor between the equation and its first derived polynomial. If there is no common divisor,...
Σελίδα 15 - Find four numbers in arithmetical progression such that the sum of the squares of the first and second shall be 29, and of the third and fourth 185.
Σελίδα 12 - AR. progression, such that the sum of the squares of the first and second be 29 ; and the sum of the squares of the third and fourth be 185.
Σελίδα 57 - ... itself, and place the product for the first term of the SECOND COLUMN. This, multiplied by the same figure, will give the first term of the THIRD COLUMN. Thus continue until the number of columns is one less than the units in the index denoting the root. Multiply the term in the LAST COLUMN by the same figure, and subtract the product from the first period, and to the remainder bring down the next period, and it will form the FIRST DIVIDEND. Again, add this same figure to the term of the FIRST...

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