17. c +23 c3+102. 18. xy+2xy - 120. 21. a2b2+4 abc-45 c. 23. (a+b)+26(a+b)+144. 38. (a2+3a)2 – 14 (a2 +3a) + 40. 39. (ab2- c2)2 - 4 b2c2. 40. x8-82x2 + 81. 41. 3(ab)- (a - b)2. 42. (x2+4)-162. 43. (x23x)2-38(x2-3x) - 80. 44. y3-3xу (xy). 45. (aa-4)2 - 4. 46. (x-4)2(3x-2) (x + 2)2. Note. Other methods for factoring will be given in Chapter XX. VIII. HIGHEST COMMON FACTOR. Note. In the present chapter we shall consider integral expressions only (Art. 65); and when we speak of one expression as exactly dividing another, it is understood that the quotient is an integral expression. 134. A Common Factor of two or more expressions is an expression which exactly divides each of them. 135. The Highest Common Factor of two or more expressions is their common factor of highest degree (Art. 24); or if there are several common factors of equally high degree, it is the one having the numerical coefficient of greatest absolute value in its term of highest degree. Note 1. The abbreviation H.C.F. is used for Highest Common Factor. Note 2. There are always two forms of the highest common factor, one of which is the negative of the other. Thus, in the expressions a2-ab and b2 - ab, either a-b or b-a will exactly divide each expression. 136. Two expressions are said to be prime to each other when unity is their highest common factor. 137. In determining the highest common factor of algebraic expressions, it is convenient to distinguish two cases. 138. CASE I. When the expressions are monomials, or polynomials which can be readily factored by inspection. 1. Required the H.C.F. of 42 a3b2, 70 a2bc, and 98 a1b3ď2. It is evident by inspection that the expression of highest degree which will exactly divide a b2, a2bc, and a1b3d2, is a2b; and by the rule of Arithmetic, the H.C.F. of 42, 70, and 98 is 14. Hence the H.C.F. of the given expressions is 14 ab. 2. Required the H.C.F. of 5xy 45xy and 10xy+40xy - 210 xy. By Arts. 128 and 129, 5x1y 45x2y = 5 x2y (x2 — 9) =5.(2+3)(–3), and 10xy+40xy-210xy 10 xу (x2+4x-21) = =10xу(x+7) (x-3). It is evident by inspection that the H.C.F. of the literal portions of the expressions is xy(x-3); and the H.C.F. of the numerical coefficients 5 and 10 is 5. Hence the H.C.F. of the given expressions is 5xy (x-3). 139. CASE II. When the expressions are polynomials which cannot be readily factored by inspection. Let A and B be two polynomials, arranged according to the descending powers of some common letter, and let the exponent of that letter in the first term of A be not lower than its exponent in the first term of B. Suppose that, when A is divided by B, the quotient is p, and the remainder C. To prove that the H.C.F. of B and C is the same as the H.C.F. of A and B. The operation of division is shown as follows: B) A (p C We will first prove that every common factor of B and C is a common factor of A and B. Let F be any common factor of B and C; and let Substituting the values of B and C from (1), we obtain A=pbF+cF= F(pb+c). (2) It is evident from (1) and (2) that F is a common factor of A and B (Art. 134). We will next prove that every common factor of A and B is a common factor of B and C. Let F' be any common factor of A and B; and let AmF and BnF". From the operation of division, we have C=A-pB. (3) Substituting the values of A and B from (3), we obtain CmF" - pn F" F" (m-pn). = (4) It is evident from (3) and (4) that F" is a common factor of B and C. It follows from the above that the H. C. F. of B and C is the same as the H.C.F. of A and B. 140. Suppose that, when B is divided by C, the quotient is q, and the remainder D; that when C is divided by D, the quotient is r, and the remainder E, and so on; and that, finally, we arrive at a remainder H, which exactly divides the preceding divisor G. By Art. 139, the H.C.F. of C and D is the same as the H.C.F. of B and C; the H. C. F. of D and E is the same as the H.C.F. of C and D; and so on. Hence the H.C.F. of G and H is the same as the H.C.F. of A and B. But since I exactly divides G, H is itself the H.C.F. of G and H. Therefore H is the H. C. F. of A and B. We derive from the above the following rule for the H. C. F. of two polynomials, A and B, arranged according to the descending powers of some common letter, the exponent of that letter in the first term of A being not lower than its exponent in the first term of B: Divide A by B. If there is a remainder, divide the divisor by it; and continue thus to make the remainder the divisor, and the preceding divisor the dividend, until there is no remainder. The last divisor is the highest common factor required. Note 1. It is important to arrange the work throughout in descending powers of some common letter; and each division should be continued until the exponent of this letter in the first term of the remainder is less than its exponent in the first term of the divisor. Note 2. If the terms of one of the expressions contain a common factor which is not a common factor of the terms of the other, it may be removed; for such a common factor can evidently form no part of the highest common factor. In like manner we may remove from the terms of any remainder any common factor which is not a common factor of the terms of the preceding divisor. Note 3. If the first term of the dividend, or of any remainder, is not divisible by the first term of the divisor, it may be made so by multiplying the dividend or remainder by any term which is not a common factor of the terms of the divisor. Note 4. If the first term of any remainder is negative, the sign of each term of the remainder may be changed. (Compare Note 2, Art. 135.) Note 5. If the terms of the given expressions have a common factor, remove it, and find the H. C. F. of the resulting expressions. The result, multiplied by the common factor, will be the H. C. F. of the given expressions. Note 6. The operation of division may usually be abridged by the use of detached coefficients (Art. 105). 1. Required the H.C.F. of 12x2 x-35 and 15x2 + 31x+10. Since 15 is not divisible by 12x2, we multiply the second expression by 4 (Note 3). |