We may here remind the student that in division, when divisor and dividend are not both positive, we perform the division just as if they were positive, and afterwards prefix to the quotient the appropriate sign given by the rule of signs. 133. To find a meaning for the symbol - α -b' we define it as a by-b; and this the quotient resulting from the division of is obtained by dividing a by b, and, by the rule of signs, prefixing +. Therefore b - α b (1). b α Again, is the quotient resulting from the division of -α by b; and this is obtained by dividing a by b, and, by the rule of signs, prefixing-. Therefore Likewise α -b is the quotient resulting from the division of a by b; and this is obtained by dividing a by b, and, by the rule of signs, prefixing These results may be enunciated as follows: (1) If the signs of both numerator and denominator of a fraction be changed, the sign of the whole fraction will be unchanged. (2) If the sign of the numerator alone be changed, the sign of the whole fraction will be changed. (3) If the sign of the denominator alone be changed, the sign of the whole fraction will be changed. The principles here involved are so useful in certain cases of reduction of fractions that we quote them in another form, which will sometimes be found more easy of application. 1. We may change the sign of every term in the numerator and denominator of a fraction without altering its value. 2. We may change the sign of a fraction by simply changing the sign of every term either in the numerator or denominator. Here it is evident that the lowest common denominator of the first two fractions is x2 - a2, therefore it will be convenient to alter the sign of the denominator in the third fraction. 20. 21. (a - b)(x-a) (b− a) (x − b) * (a - b)(x-a) (ba) (b-x)" 2a+y + x+y-a (x − a) (a − b) † (x —b) (b − a) ̄ ̄ (x − α) (x − b)* 1 - 1 1 (a2 — b2) (x2+b2) * (b2 – a2) (x2+a2) ̄ ̄ (x2+a2) (x2+b2)* 24. 26. 4a3 (a+x) ̄ ̄ ̄ 4a3 (x − a) † 2a2 (a2+x2) ̄ ̄ a3 − x8° --- y 23+33 x2-y2 x2+y2 y⋆. x1 ' (x+y) (x2+ y2) a (a2 _b2) + b(a2 + b2) + ab (b* — a3) ̄ b3_a3• 1 + 134. From Art. 133 it follows that, (1) Changing the signs of an odd number of factors of numerator or denominator changes the sign before the fraction. (2) Changing the signs of an even number of factors of numerator or denominator does not change the sign before the fraction. + 1 (a−b)(a−c) * (b−c)(b−a)† (c−a ) (c —b) By changing the sign of the second factor of each denominator, we obtain Now it is readily seen that the L. C. M. of the denominators is (a−b)(b−c)(c—a), and the expression 135. There is a peculiarity in the arrangement of this example which it is desirable to notice. In the expression (1) the letters occur in what is known as Cyclic Order; that is, b follows a, a follows c, c follows b. Thus if a, b, c are arranged round the circumference of a circle, as in the annexed diagram, if we start from any letter and move round in the direction of the arrows, the other letters follow in cyclic order, namely abc, bca, cab. a The observance of this principle is especially important in a large class of examples in which the differences of three letters are involved. Thus we are observing cyclic order when we write b-c, c-a, a−b; whereas we are violating cyclic order by the use of arrangements such as b—c, a—c, a−b, or -c, b-a, b-c. It will always be found that the work is rendered shorter and easier by following cyclic order from the beginning, and adhering to it throughout the question. a EXAMPLES XV. h. Find the value of α 1. (a−b) (a−c) (a − b ) ( a − c) + (b −c) (b − a) + (c− a) (c—b) • α (a−b) (a–c) + (b−c) (b− a) + (e—a) (c—b) • + + (x − y) (x − z) † ( y − 2) (y − x) TM (z — x) (z − y) ̊ b-c (a − b) (a − c) † (b−c) (b− a) * (c− a) (c — b) * (x − y) (x − z) † ( y − 2) ( y − x) ̄ (z − x) (z − y) * (a-b) (a-c) (b-c) (b-a) (c-a) (c-b)* 2. 3. 4. 5. 6. 7. 8. 9. 10. (x-y)(x-2) (y-z) (y-x) (z-x)(z-y)* + (p − q) (p − r) + (1 − r) (1 − p) † (r− p) (r − q) * r+p-q p+q-r a2 (a2-b2)(a2-c2) + (b2 - c2) (b2 — a2)* (c2-a2) (c2-b2) * x+y + x + y |