11. Find the L. C. M. of x2 - 4a3, x3 +2ax2 + 4a2x + 8a3 and x3- 2αx2 + 4a2x - 8α3. 12. Find the L. C. M. of x-(a+b)x+ab, x2 - (b+c) x + be and x-(c+a) x + ca. 13. Required the L. C. M. of 2x2+(2a-3b) x2 - (2b+3ab) x + 363 and 2x-(3b-2c) x-3bc. 14. Required the L. C. M. of 6 (a3 — b3) (a − b.)3, 9 (a1 — b1) (a - b)2 and 12 (a3 — b3)3. VIII. FRACTIONS. 131. We propose to recall to the student's attention some propositions respecting fractions which he has already found in Arithmetic, and then to shew that these propositions hold universally in Algebra. In the following articles the letters represent whole numbers, unless it is stated otherwise. a b 132. By the expression we indicate that a unit has been divided into b equal parts, and that a of such parts are taken. Here α b is called a fraction; a is the numerator and b the denominator, so that the denominator indicates into how many equal parts the unit is to be divided, and the numerator indicates how many of those parts are to be taken. Every integer may be considered as a fraction with unity for its denominator; that is, p = 133. Rule for multiplying a fraction by an integer. Either multiply the numerator by that integer, or divide the denominator by that integer. α Let denote any fraction, and c any integer; then will a ac For in each of the fractions and the unit is divided into b equal parts, and c times as many parts are taken α Again; let denote any fraction, and c any integer; then bc α α For in each of the fractions and the same bc 134. Rule for dividing a fraction by an integer. Either multiply the denominator by that integer, or divide the numerator by that integer. a b a α Let denote any fraction, and c any integer; then will 1 * is th of 8. bc с This demonstrates the first form of the Rule. denote any fraction, and c any integer; then ac For is c times, by Art. 133; and thereb ac fore I is lth of 땅. b This demonstrates the second form of the Rule. 135. If any quantity be both multiplied and divided by the same number its value is not altered. Hence if the numerator and denominator of a fraction be multiplied by the same number t value of the fraction is not altered. For the fraction is multiplied by any number by multiplying its numerator by that number, and is divided by the same number by multiplying its denominator by that number. (Arts. 133 and 134.) Thus a ac And so also if the numerator and denominator of a bbc fraction be divided by the same number the value of the fraction is not altered. 136. Hence, an Algebraical fraction may be reduced to another of equal value by dividing both numerator and denominator by any common measure; when both numerator and denominator are divided by their G. C. M. the fraction is said to be reduced to its 6x2-7x-20 lowest terms. For example, consider the fraction 4x-27x+5 Here the G. C. M. of the numerator and denominator will be found to be 2x-5; hence, dividing both numerator and denominator by this we obtain change the signs of the numerator and denominator of a fraction without altering the value of the fraction. 138. To reduce fractions to a common denominator:-multiply the numerator of each fraction by all the denominators except its own for the numerator corresponding to that fraction, and multiply all the denominators together for the common denominator. ebd bdf thus adf cbf and ƒ ̄ bdƒ3 bdf' bdf' are fractions of the same value respectively as the proposed fractions, and having the common denominator bdf. 139. If the denominators have any factors in common, we may proceed thus:-find the L. C. M. of the denominators and use this as the common denominator; then for the new numerator corresponding to each of the proposed fractions, multiply the numerator of that fraction by the quotient which is obtained by dividing the L. C. M. by the denominator of that fraction. Thus suppose, for example, that the proposed fractions are α b Here the L. C. M. of the denominators is mxyz ; mx' my' and C mz 140. To add or subtract fractions,―reduce them to a common denominator, then add or subtract the numerators and retain the common denominator. |