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. 1897 Excerpt: ...If 26. 1.0064. 18. 12.5. 21. 0.9. 24.-fa. 27. 36. CUBE ROOT. 193. Cube Root of Polynomials. Our object is to determine such a relation between the terms of a binomial, or, in general, of a polynomial, and the terms of its cube (as between a + b, and its cube, a3 + 3a36 + Sab + &'), that we may be able to state this ...
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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. 1897 Excerpt: ...If 26. 1.0064. 18. 12.5. 21. 0.9. 24.-fa. 27. 36. CUBE ROOT. 193. Cube Root of Polynomials. Our object is to determine such a relation between the terms of a binomial, or, in general, of a polynomial, and the terms of its cube (as between a + b, and its cube, a3 + 3a36 + Sab + &'), that we may be able to state this relation in the inverse form as a general method for determining the cube root of any polynomial which is a perfect cube. a" + 3a26 + 3aZ2 + 63 a + b The first term of the root, a, is the cube root of the first term, a3, of the cube expression. The second term of the root, b, occurs in the second term of the cube expression, 3a26, and may be obtained from it by dividing it by 3a2; that is, by three times the square of the first term of the root (called the trial divisor). If we take the trial divisor, and add to it three times the product of the first term of the root by the second term, Zab, and also the square of the second term of the root, 62, we get 3a2 + Zab + b2 (called the complete divisor); this multiplied by the second term of the root gives 3a26 + 3ab2 + 63, the rest of the cube expression after a3 has been subtracted. This last step, therefore, furnishes a test of the accuracy of the work. 194. Three or More Terms in the Boot. In cubing a trinomial, a + b + c, we may regard a + b as a single quantity, and denote it by p, and obtain the cube in the form p + 3p2c + 3pc2 + c Evidently we may reverse this process, and extract a cube root to three terms, by regarding two terms of the root when found as a single quantity. So a fourth term or any number of terms of a root may be found by regarding, in each case, the root already found as a single quantity. We will now extract the cube root of a polynomial expression indicating at ea...
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