## A catechism and notes upon the algebras of Bourdon and Lacroix: for the use of the students of the New-York University |

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BOURDON AND LACROIX cient coeffi coefficients cube root denominator difference dividend entire function entire polynomial equation containing equation found equation resolved EXTRACT THE nth final equation formula give the number given equation given number given polynomial greatest number greatest relative common hand significant figures highest exponent horizontal row imaginary roots independent number last term least limit less letter of arrangement logarithm monomial extracted multi multiplication of monomials n—1 power Naperien negative roots Newton's method nth power nth root number greater number of rows number of units operations indicated partial products perfect square plied positive roots principal letter principle quantity equal quotient rational and entire real roots odd relative common divisor relative divisor remainder right hand significant right to left root found rule depend rule for extracting rule for finding second degree second member second term square root subtracting the nth subtractive terms Suppose trinomial unknown quantity whole number

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Page 24 - There will be as many figures in the root as there are periods in the given number.

Page 26 - Subtract its power from the given polynomial, and divide the first term of the remainder by n times the (n— 1) power of this root ; the quotient will be the second term of the root.

Page 19 - The first method which suggests itself is one similar to that usually given to determine the coefficients of the equation whose roots are the squares of the differences of the roots of any given equation.

Page 9 - ... the first root is essentially positive. The second root, being equal to p, minus a quantity less than p, is essentially positive. Hence, in the fourth form, both roots are positive. See problems 14 and 15, Art. 212. REVIEW. — 215. Show that every quadratic equation has two roots, and only two. 216. To what is the sum of the roots equal? To what is the product equal?

Page 6 - REVIEW. — 80. To what is the square of the sum of two quantities equal ? 81.

Page 7 - What is the greatest common divisor of 918 and 5221 (251.) In the application of this rule to polynomials, some modification may become necessary. It may happen that the first term of the dividend is not divisible by the first term of the divisor. This may arise from the presence of a factor in the divisor which is not found in the dividend, and may therefore be suppressed. For, since the greatest common divisor of two quantities is only the product of their common factors, it cannot be affected...

Page 26 - If we continue the operation, we shall find that the first term of any new remainder, divided by n times the (n — 1)'* power of the first term of the root, will give a new term of the root.

Page 2 - The numbers over the questions are the numbers of the articles of Bourdon in which the answers are to be found ; and the numbers before them are those of the corresponding articles in Lacroix. Where the book does not contain the answer to the question, no number is placed.

Page 1 - PREFACE. THE French Algebras, so superior as books of instruction, are written in the form of a continuous treatise, without distinct divisions. The student is in consequence often at a loss to know what is to be carefully retained in the memory, and what may 3' be passed over more rapidly, as being intended only < for immediate explanation.

Page 5 - ART. 138. THEOREM. — If the same quantity be added to both terms of a proper fraction, the new fraction resulting will be greater than the first; but if the same quantity be added to both terms of an improper fraction, the new fraction resulting Witt be less than the first, Let - be a proper fraction, a being less than b. b Let m represent the quantity to be added to each term, then...