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. 1844 Excerpt: ...the general problem may be stated thus: a given polynomial, denoted by d, is the product of another given polynomial a', by an unknown polynomial q; it is required to find q. Since d=df x g; if q is known, d can be reproduced by multiplying the terms of d? successively by j, and forming the algebraic sum of the partial ...
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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. 1844 Excerpt: ...the general problem may be stated thus: a given polynomial, denoted by d, is the product of another given polynomial a', by an unknown polynomial q; it is required to find q. Since d=df x g; if q is known, d can be reproduced by multiplying the terms of d? successively by j, and forming the algebraic sum of the partial products. Of these partial products the like are reduced with each other, but such as are unlike any of the others are found in d (or d'x.q) uncombined with other quantities, and capable of resolution into the very terms of d' and q of which they are the product. Now, the partial product which arises from the multiplication of that term of d' which is affected with the highest exponent of one of the letters by that term of q which is affected with the highest exponent of the same letter cannot be reduced with any of the other partial products (Art. 12'). This partial product must, therefore, be itself a term of d, that affected with the highest power of the same letter. Whence, if that term of d which is affected with the highest exponent of one of the letters is divided by that term of d' which is affected with the highest exponent of the same letter, the result must be that term of q which is affected with the highest exponent of that letter. Since the dividend d is obtained by adding together the partial products of all the terms of d' by each term of 9, it follows that if (f is multiplied by that term of q which has been found, and the product subtracted from d, the remainder must be equal to the product of rf' by the unknown terms of q. The same reasoning may be used in the case of this remainder and d' as has been employed in the case of d and rf', and the same conclusion deduced from it, namely, that if that term of the remainder or ne...
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