EXERCISES Find the H.C.F. of the following: 1. 2x4 + 2 x3 — x2 - 2 x − 1 and x + x3 + 4x + 4. This process may be performed in the following more compact form. 5. x3 2 x2 15x+36 and 3x2 - 4 x 15. 6. x4 3 x3 + x2 + 3x - 2 and 4x3 – 9x2 + 2x + 3. 7. 4x3- 18 x2 + 19x 3 and 2x4 - 12 x3 19 x2 - 6x + 9. 8. x4 + 4x3- 22 x2 - 4 x + 21 and x4 + 10 x3 + 20 x2 9. 6a4x3- 9 a3x2y — 10 a2 xy2 + 15 ay3 and 10 a5x6у2 – 15 aax5y3 + 8 a3x1y1 - 12 a2x3y5. 40. Least common multiple. The least common multiple of two or more polynomials is the polynomial of least degree that contains them as factors. We may find the least common multiple of several polynomials by the following RULE. Multiply together all the factors of the various polynomials, each factor having the greatest exponent with which it appears in any of the polynomials. 41. Second rule for finding the least common multiple. The previous rule is evidently equivalent to the following RULE. Multiply the polynomials together and divide the product by their highest common factor. 8. x 1, 2 x2 9. x8 5 х 3, and 2x37x2 + 2x + 3. 9x226x-24 and x3 - 10x2 + 31x - 30. 10. 2x2 - 3x + 9, x2 — 6x + 9, and 3x2 9xbx+3b. CHAPTER III FRACTIONS 42. General principles. The symbolic statements of the rules for the addition, subtraction, multiplication, and division of algebraic fractions are the same as the statements of the corresponding operations on numerical fractions given in (2), (3), and (4), § 6. This is immediately evident if we keep in mind the fact that algebraic expressions are symbols for numbers and that if the letters are replaced by numbers, the algebraic fraction becomes a numerical fraction. 43. PRINCIPLE I. Both numerator and denominator of a fraction may be multiplied (or divided) by the same expression without changing the value of the fraction. This follows from (5), § 6. 44. PRINCIPLE II. If the signs of both numerator and denominator of a fraction be changed, the sign of the fraction remains unchanged. This follows from Principle I, when we multiply both numerator and denominator by -1. 45. PRINCIPLE III. If the sign of either numerator or denominator (but not both) be changed, the sign of the fraction is changed. This follows from (6), § 6. 46. Reduction. A fraction is said to be reduced to its lowest terms when its numerator and denominator have no common factor. We effect this reduction by the following RULE. Divide both numerator and denominator by their highest common factor. 47. Least common denominator of several fractions. We have the following RULE. Find the least common multiple of the various denominators. Multiply both numerator and denominator of each fraction by the expression which will make the new denominator the least common multiple of the denominators. Solution: The L.C.M. of the denominators is x (4x2 - 1). Thus the frac 48. Addition of fractions. This operation we perform as follows: RULE. Reduce the fractions to be added to their least common denominator. Add the numerators for the numerator of the sum, and take the least common denominator for its denominator. 49. Subtraction of fractions. This operation we perform as follows: RULE. Reduce the fractions to their least common denominator. Subtract the numerator of the subtrahend from that of the minuend for the numerator of the result, and take the least common denominator for its denominator. 50. Multiplication of fractions. This operation we perform as follows: RULE. Multiply the numerators together for the numerator of the product, and the denominators for its denominator. 51. Division of fractions. This operation we perform as follows: RULE. Invert the terms of the divisor and multiply by the dividend. REMARK. Since a fraction is a means of indicating division, are two expressions for the same thing. a |