From the number of terms in the quotient already obtained in the above example, the learner will readily infer a law, by which the quotient may be continued at pleasure without performing any more operations. 53. Miscellaneous examples in the division of algebraic quantities. 1. To divide 32 x6-8 c2 x3 + 12 x3-c2+1 by 4x3-c2+1. 2. To divide x5 + 1 by x + 1. 3. To divide 1-5 x 10 x2-10 x3 + 5 x1 — x5 by 1-2x+x2. 4. To divide a5 + x5 by a +x. 5. To divide a5-5a4 z+10 a3 22-10 a2 23 +5 a z1-25 by a2-2azz2. 6. To divide 6 x6 - 5 x5 y2+21 x3 y3—6 x1 y1+x2 y5 + 15 y by 2 x3-3x2 y2+5 y3. 7. To divide 1 by 1- -α. 8. To divide 1+ ax + bx2+c x3 + dx2 + &c., by 1 -X. Ans. 1+1+1 | x2 &c. 9. To divide a bx + cx2-dx3+ &c., by 1+x. b 54. As a further illustration of the use of algebraic symbols, art. 29, the following propositions may now be demonstrated. 1. If the sum of any two quantities be added to their dif ference, the sum will be twice the greater. 2. If the difference of any two quantities be taken from their sum, the remainder will be twice the less. 3. The second power of the sum of two quantities contains the second power of the first quantity, plus double the product of the first by the second, plus the second power of the second. 4. The second power of the difference of two quantities is composed of the second power of the first quantity, minus the double product of the first by the second, plus the second power of the second. 5. The product of the sum and difference of two quantities is equal to the difference of their second powers. 55. When the division of two algebraic quantities cannot be exactly performed, the quotient, as we have seen, is expressed in the form of a fraction, the dividend being taken for the numerator and the divisor for the denominator. A fraction in algebra has the same signification as a fraction in arithmetic; the denominator shows into how many parts unity is divided, and the numerator how many of these parts α are taken. Thus, in the algebraic fraction, unity is suppos ed to be divided into b parts, and a number a of these parts is supposed to be taken. REDUCTION OF FRACTIONS TO THEIR LOWEST TERMS. 56. A fraction is said to be in its lowest terms, when there is no quantity, that will divide both of its terms without a remainder. To reduce a fraction therefore to this state, we suppress in the numerator and denominator the factors, which are common to them. When the two terms of an algebraic fraction are simple quantities, it will be easy, from inspection, to determine the factors common to them; but if the terms of the fraction are polynomials, this will not be so easy, and we must in this case have recourse to the method of the greatest common divisor. By the greatest common divisor of two algebraic quantities we understand the greatest in regard to coefficients and exponents, that will exactly divide these quantities. Its theory rests upon the same two principles, as that of the greatest common divisor in arithmetic, viz. 1o. The greatest divisor common to two quantities contains as factors all the particular divisors common to these quantities and no others. 2°. The greatest divisor common to two quantities is the same with the greatest divisor common to the less of these quantities and the remainder after the division of the greater by the less. 57. This being premised, let it be proposed to find the greatest common divisor of the polynomials a3 — a2 b + 3 a b2 - 3 b3, and a2. ·5 a b + 4 b2 Pursuing the same general course as in arithmetic, we commence by dividing the first of the proposed polynomials by the second; we thus obtain a 4b for a quotient with a remainder 19 a b2 19 b3. By the second of the above principles the question is now reduced to finding the greatest common divisor to this remainder and the divisor a2-5 ab+462. But 19 a b2 - 19 b3 may be put under the form 19 b2 (a - b); and since the factor 19 62 of this quantity will not divide a2-5ab4b2, it will not from the first of the above principles be a factor of the greatest common divisor sought, it may therefore be rejected; thus the question will be still further reduced to finding the greatest common divisor to a b and a2 5 ab+4b2 Dividing the last of these two quantities by the first we obtain an exact quotient a 4b; whence a b is their greatest common divisor; and by consequence it is the greatest common divisor of the polynomials proposed. The following is a table of the calculations. 1st operation a3 a b+3ab2 a3-5a2 b+4ab2 5ab4b2 a+46 363 Let it be proposed, as a second example, to find the greatest common divisor of the polynomials 5 b3 — 18 b2 a + 11 b a2 — 6 a3, and 7 b2 — 23 b a + 6 a2 In this example 5 b3, the first term of the dividend, is not divisible by 7 b2, the first term of the divisor. It will be observed, however, that 7, the coefficient of the first term of the divisor, will not divide the remaining terms of the divisor. We may therefore, in virtue of the first principle, multiply the dividend by 7 without affecting the greatest common divisor sought. Performing this operation, we have for the dividend. 35 b3-126 b2a77 b a2-42 a3 Dividing next 35 b3 by 7 b2, we obtain 5 b for a quotient. Multiplying the whole divisor by 5 b, and subtracting, we have for a remainder - 11 b2 a + 47 b a2 42 a3. The exponent of b in this remainder, being equal to the exponent of the same letter in the divisor, we continue the operation; and in order to render the first term divisible by the first term of the divisor, we multiply anew by 7, which gives - 77 b2 a + 329 b a2 294 a3. Dividing this by the divisor, the quotient is 11 a, which we separate from the other by a comma, to show that it has no connexion with it, and the remainder is 76 b a2. 228 a3, or 76 a3 (b −3 a). Suppressing, as in the preceding example, the factor 76 a2, the question is reduced to finding the greatest divisor common to b-3 a and 7 b2-23 ba+6 a2. Dividing, therefore, the last of these quantities by the first, we obtain an exact quotient 7b-2 a; whence b-3 a is the greatest common divisor sought. The suppression of the factors 19 b2, 76 a2 respectively in the first remainder of the preceding examples serves not only to simplify the calculations, but it is also indispensable; for unless this is done, we must multiply in the first example the new dividend by 19 b2, and in the second by 76 a2, in order to render the first term divisible by the first term of the divisor; we should thus introduce into this dividend a factor, which is found in the divisor, and by consequence we should introduce into the greatest common divisor sought a factor, which does not belong to it. The suppression of a factor is the same, it is evident, as dividing the proposed by the factor, required to be suppressed. 3. To find the greatest common divisor of the polynomials a3 +5 a2 b + 3 a b2 — b3 4. To find the greatest common divisor of the polynomials 5. To find the greatest common divisor of the polynomials a3-10 a b2+363 6. To find the greatest common divisor of the polynomials 58. Let it be proposed next to find the greatest common divisor of the polynomials and 15 a5+ 10 a* b + 4 a3 b2 + 6 a2 b3 — 3 a ba · 12 a3 b2 + 38 a2 b3 + 16 a ba ——— 10 b3. Before proceeding to the division of the proposed polynomials, we observe that the first contains the letter a as a factor |