The first three terms being the square of a + b, we change the expression to (a+b)2 - c2. But this last expression is the difference of two squares; and hence we may substitute for it, the product of the sum and difference, and we obtain, (a+b)2-c2= (a+b+c)(a+b-c). For a third example of the mutations which an algebraic quantity may be made to undergo, take the following, which becomes successively, In like manner, a2-b2 - x2 + 2bx, may be written, (20.) Since algebraic fractions represent unexecuted divisions, and are in all respects similar to arithmetical fractions, the same rules will apply to both. REDUCTION OF FRACTIONS. To reduce fractions to their lowest terms. RULE. Divide both terms of the fraction by any quantity that will divide them without remainder. Divide these quotients again, in like manner, till it appears that no quantity can be found which will exactly divide both; or, which is the same thing, suppress all the factors common to both terms of the fraction. Here 7 abc is the common divisor, and dividing according to the Decomposing each term of this fraction into its factors, we have, 10ax(a2 + 2ax + x2) α(α2 - x2) 10ax(a+x) (α + x). and suppressing factors common to both, 10x(α + x) ; α(α-x) (a+x) 10ax + 10x2 , or -, the fraction in its lowest terms. 5a5b2 - 20ab6 a*b2+2a2b+ = a*b2 + 2ab+ L 5ab2(a2-464) 5(a2 - 262) (a2 +262) a2b2(a2 + 2b2) 5(a2-26), the answer. , 1 a(a2+262) 10. Reduce a2+2a2cx2+c2x Ans. a2 +cx2 Басх It is sometimes the case, that the terms of the fraction are so complicated, that it is not possible to reduce them by the preceding rule, and it then becomes necessary to find their greatest common divisor, and divide each term of the fraction by it, in order to effect the reduction. To find the greatest common divisor of two polynomials. RULE. (21.) 1st. Arrange the polynomials with reference to the powers of the same letter. 2d. Divide the polynomial, which has the highest exponent of this letter, by the other; and if there be a remainder, divide the divisor by this remainder, and continue the operation in the same manner, always dividing the last divisor by the last remainder, till the division can be executed without remainder; the last divisor is the greatest common divisor of the two polynomials. NOTE. If, in the course of the operation, it happens at any time, that any factor is contained in all the terms of one polynomial, and not in all the terms of the other; this factor may be expunged from the polynomial containing it; for, since it divides one of the polynomials, and will not divide the other, it cannot be a factor of their common divisor. EXAMPLES. 1. Find the greatest common divisor of as - ax, and a3 + a2x -ах2 - x3. In this example, a5 - ax contains a in all its terms, and the other does not: a therefore cannot be a factor of the common divisor; for if it were, it must divide a3 + a2x-ax2-x3; we can therefore expunge it from the first; and the problem is, to find the greatest common divisor of a* - r, and a3 + a2x - ах2 -х3. The division can be carried no farther, since the first term of the divisor is not contained in any term of the dividend; we therefore take the divisor for the dividend, and the remainder for a new divisor, expunging from it the factor 2x2, which is not found in all the terms of the new dividend. a2 - x2 is therefore the greatest common divisor. 2. Find the greatest common divisor of 12a +12a2b2, and 8a3b2 24a2b3 +24ab4 - 865. 24a3b Expunging factors not common to both, these quantities become a2-2ab + b2, and a3-3a2b+3ab2-b2. a2-2ab+b2 is therefore the greatest common divisor. NOTE 2.-If the letter with respect to which the quantities are arranged, have the highest exponent in the dividend, and the coefficient of the first term of the dividend be not divisible by the coefficient of the first term of the divisor, the dividend must be multiplied by such a number as will make this coefficient divisible. This will not affect the correctness of the result, for the factor thus introduced into the dividend, and not into the divisor, can form no part of the greatest common divisor. 3. Find the greatest common divisor of 3a +5a2b2-5ab3 +264, and 6a3 + 8a2b-11 ab2 + 2b3. 1st division. 5a3b 2×3a2 - 5a2b+5a2b2-5ab2+2b46a2+8a2b--11ab2+2b3 6a--10a3b+10a2b2-10ab3+4b4 a-3b 6a2+8a3b-11a2b2+2ab3 18a2b+21a2b2 - 12ab3 +464 18a3b24a2b2 + 33ab2-664 45a2b2 --45ab3 + 1064, or 5b2(9a2--9ab +262) 2d division. 3 × 6a3+8a2b 11ab2 + 2b39a2- 9ab + 262 18a3 +24a2b-33ab2 + 6b3 ) 18a3-18a2b+ 4ab2 12a +14b 3a-2b, the last divisor, is the greatest common divisor required. We now propose to find the common divisor of these same polynomials by arranging them with reference to the powers of b. |