250. Two or more expressions may have several common factors. Thus, 35 x3y2, 21 x2ys and 42 x3ys have what common factors of the first degree? of the second degree? of the third degree? of the fourth degree? Can you find a common factor of these expressions of higher degree than the fourth? 251. The highest common factor (H. C. F.) of two or more monomials is the greatest common divisor of their numerical coefficients multiplied by their highest degree literal common factor. Thus, 7 x2y2 is the H. C. F. of 35 x3y2, 21 x2y3, and 42 x3y3. 252. The H. C. F. in algebra corresponds to the greatest common divisor (G. C. D.) in arithmetic. The G. C. D. is the largest number that will exactly divide two or more numbers; the H. C. F. is the highest degree algebraic expression that will divide two or more expressions. We may find the G. C. D. of 12, 18, 24 by factoring thus: 18 = 2.32, 24 = 23.3. 12 = 22.3, Similarly, we may find the H. C. F. of two or more algebraic expressions. Find the H. C. F. of 12 a2bc, 18 a3b2c2, 24 a3c. SOLUTION. 12 a2bc22. 3. a2bc. The H. C. F. is the G. C. D. of the numerical coefficients, 6, multiplied by their highest degree literal common factor a2c; that is, the H. C. F. is 6 a2c. 253. To find the H. C. F. of two or more algebraic expressions, multiply together the lowest powers of all the prime factors common to all the expressions. In the case of monomials the H. C. F. is seen by inspection. If any of the expressions are polynomials, factor them into prime factors. 25. 2a2b+2 ab2 — 2 abc, 3 bc2 — 3 b2c — 3 abc. 26. a2+b2 c2+2 ab, a2 - b2 + c2 +2 ac. 255. A product is a multiple of any of its factors. Thus, 3 x2y is a multiple of x; of xy; of 3x; etc. 256. A common multiple of two or more expressions is a multiple of each of them. Thus, 6 x2y3 is a common multiple of 3 x, 2 y and xy. Two or more expressions have always an infinite number of common multiples. Thus, 3x, 2y, and xy have as common multiples 6 xy, 6 x2y2, 6 x2y, 12 xy, etc., indefinitely. Can you find a common multiple of these three monomials of lower degree than the second? 257. The lowest common multiple (L. C. M.) of two or more monomials is the arithmetical least common multiple of their numerical coefficients multiplied by their lowest degree literal common multiple. 258. In arithmetic the least common multiple of two or more numbers is the smallest number which may be exactly divided by each of them. In algebra the L. C. M. of two or more expressions is the lowest degree expression which may be exactly divided by each of them. 259. To find the L. C. M. of two or more algebraic expressions, multiply together the highest powers of all the different prime factors in the expressions. The L. C. M. of monomials is seen by inspection. If the expressions are polynomials, first factor them into prime factors. 1. Find the L. C. M. of 9 b3c, 12 ac2, 4 abc3. SOLUTION. 9 b3c = 32b3c, 12 ac2 3. 22ac2, 4 abc3 = 22abc3. ... L. C. M. = 32. 22ab3c3 or 36 ab3ç3. 2. Find the L. C. M. of a2-3 a +2, a2-1, a2 - 4 a +4. 260. Find the L. C. M. in each of the following, leaving the results, in the case of polynomials, in factored form: 11. a3a, a2 - 2a+1, 2 a25a+3. 12. 2 a2 5 a+ 3, 4 a2 - 13 a +3, 8 a2 - 6a+ 1. 13. What is the L. C. M. of two expressions that have no common factor? 14. 2x2+x 1, x2-x-2, 2 x2 − 5 x + 2. 15. 2(2x+5), 3x+6, 2x2 + 9 x + 10. 16. x2 + 3x + 2, x2 + 4 x + 3, x2 + 5 x + 6. 20. 2 2x a + ax, 3-3 x − b + bx. 21. 3 a25 ax + 2 x2, 4 a2 - 9 ax + 5 x2. X. FRACTIONS 261. An algebraic fraction is an indicated division. Thus, (read the fraction, a divided by b) is the indicated quotient of a divided by b. The numerator of the fraction is the dividend and the denominator is the divisor. Terms of a Fraction. The numerator and the denominator are the terms of a fraction. The denominator of a fraction cannot be 0 since dividing by 0 has no meaning in the ordinary sense of division. The topics studied under fractions in algebra agree closely with those of arithmetic, and the methods are similar. EXERCISE 262. 1. Reduce 1 to lowest terms. Also . 2. Change the improper fraction 2 to a mixed number. Give the rule. 3. Change,, to equivalent fractions having the least common denominator. Give the rule. 4. State the rule for adding arithmetical fractions. |