After the first division the factor a is removed as explained in Art. 150; then the factor 5 is introduced because the first term of 4-7a-19a2-8a33 is not divisible by the first term of 5-7a – 12a2. At the next stage a factor -5 is introduced, and finally the factor 284a2 is removed. 152. From the last two examples it appears that we may multiply or divide either of the given expressions, or any of the remainders which occur in the course of the work, by any factor which does not divide both of the given expressions. EXAMPLES XVIII. b. Find the highest common factor of 1. 2x+3x2+x+6, 2x2+x2+2x+3. 2. 2y3-9y2+9y-7, y3 - 5y2+5y −4. 3. 2x3+8x2-5x-20, 6x3- 4x2 - 15x+10. 4. a3+3a2-16a+ 12, a3 + a2 – 10a +8. 5. 6x3-x2-7x-2, 2x3-7x2+x+6. 6. q3-3q+2, q3-5q2+7q-3. 7. a1+a3-2a2 + a −3, 5a3+3a2 − 17a + 6. 8. 3y4-3y3-15y2-9y, 4y5-16y4 - 44y3 - 24y2. 9. 15x1-15x3+10x2 – 10x, 30x5 +120x1 +20x3 +80x2. 10. 2m++7m3 +10m2+35m, 4m2 + 14m3 – 4m2 – 6m+28, 11. 3x4 -923+12x2 - 12x, 6x3- 6x2 - 15x+6. 12. 2a5-4a4-6a, a5+a1-3a3-3a2. 13. x3+4x2-2x-15, x3-21x - 36. 14. 9a1+2a2x2+x4, 3a1 - 8a3x+5a2x2 - 2ax3. 15. 2-3a+5a2 - 2a3, 2-5a+Sa2 - 3a3. 16. 3x2-5x3 – 15x1 – 4x3, 6x - 7x2 - 29x3- 12xa. [For additional examples see Elementary Algebra.] CHAPTER XIX. FRACTIONS. 153. THE principles explained in Chapter XVIII. may now be applied to the reduction and simplification of fractions. Reduction to Lowest Terms. 154. Rule. The value of a fraction is not altered if we multiply or divide the numerator and denominator by the same quantity. An algebraical fraction may therefore be reduced to an equivalent fraction by dividing numerator and denominator by any common factor; if this factor be the highest common factor, the resulting fraction is said to be in its lowest terms. Note. The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary. 155. When the factors of the numerator and denominator cannot be determined by inspection, the fraction may be reduced to its lowest terms by dividing both numerator and denominator by the highest common factor, which may be found by the rules given in Chap. XVIII. Example. Reduce to lowest terms 3x3- 13x2+23x - 21 The H.C.F. of numerator and denominator is 3x - 7. Dividing numerator and denominator by 3x-7, we obtain as respective quotients x2 -2x+3 and 5x2 - x - 3. Thus 3x3- 13x2 + 23x-21 (3x-7)(x2 - 2x+3) x2-2x+3 = 156. If either numerator or denominator can readily be resolved into factors we may use the following method. Example. Reduce to lowest terms x3+3x2-4x 7x3-18x2+6x+5 The numerator = x(x2+3x-4) = x(x+4)(x − 1). Of these factors the only one which can be a common divisor is x-1. Hence, arranging the denominator so as to shew x-1 as a factor, the fraction = x(x+4)(x − 1) = x(x+4) (x-1)(7x2-11x-5)-7x2 - 11x- 5 Multiplication and Division of Fractions. 157. Rule. To multiply together two or more fractions: multiply the numerators for a new numerator, and the denominators for a new denominator. and so for any number of fractions. In practice the application of this rule is modified by removing in the course of the work factors which are common to numerator and denominator. by cancelling those factors which are common to both numerator and denominator. 158. Rule. To divide one fraction by another: invert the divisor, and proceed as in multiplication. 17. X 18. x2+2x-48 2x2-15x+18 (x-6)2 a2+8ab-9b2a2 - 7ab+12b2 a3 + a2b ab2 a2+6ab-2762 ax2 - 16a3 x2+ax - 20a2 x2 ax-30a3 ax2+9a2x+20a3 x2+8αx+15a2 (a - b)2 - c2 a2 + ab + ac (a+b)2 - c2 X a2 - ab + ac (a−c)2-b2 (a+b+c)2' [For additional examples see Elementary Algebra.] X a3 - b3 a2-3ab-4b2" x2-8ax+16a2 ÷ |