14. Show that (a + wẞ + w2y) (a + w2ß + wy) = Za2 15. Show that (x + wy + w2z)3 + (x + w2y + wz)3 =22x3-3x2y + 12 xyz. 16. Distribute (x + wy + w2z) (x + w2y + wz) (x + y + z). 17. Show that (x + wy + w2z)3 − (x + w2y + wz)3 = √3 (x − y) (y − z) (z − x). 18. Find the relation between a and b when (a + b)x2 − (a2 — b2)x + a2b + ab2 is a complete square. 58. The integral function of x, factors into x4-10x3 +35 x2 - 50x + 24, (x-1) (x-2) (x − 3) (x − 4). The numbers 1, 2, 3, 4 are the roots of the function, because from these, and the variable x, we may build up the function by multiplication, or, so to speak, cause it to grow up. If any of these roots be put for x, the substitution will cause the function to vanish, since it makes one of the factors zero. And, conversely, the only single substitution that will make the function vanish must make one of the factors vanish, or is the substitution of one of the roots for x. Hence the roots of an integral function of any variable are those quantities which, when put for the variable in the function, cause it to vanish. And reciprocally any quantity, which put for the variable will cause the function to vanish, is a root. Thus the roots of x3 + x2a + xΣab + abc are (x + α)(x+b)(x+c). 59. The expression - 10+ 35 x2-50 x + 240 is a conditional equation, or simply an equation in which x is to have such a value as will make the expression an identity (Art. 22). We have seen, in the preceding article, that this will be effected by making x equal to any one of the roots of the function, namely, 1, 2, 3, or 4. It is readily seen that the same principle applies to functions of any degree. Hence: (1) In the equation formed by putting an integral function of a variable equal to zero, we obtain the roots of the equation by separating the function into factors linear in the variable. The determination of any one of these factors is a solution of the equation, and the determination of all these factors is the complete solution. (2) The whole number of solutions, or the number of roots which the equation has, is the number of linear factors into which the function is theoretically separable, and this is the same as the degree of the function in the variable. The solution of an equation is thus equivalent to the factorization of the function into factors linear in the variable. 60. When the roots of an integral function or of the corresponding equation are all real and all rational, they can generally be found. Also, the methods of factoring now at our disposal are sufficient for the linear factorization of all integral functions of a single variable of not more than two dimensions; but these methods are not sufficient for the general factorization of functions of more than two dimensions. They suffice, however, for many special and particular cases. Ex. 1. To find all the solutions of x4 x3- 2x22 + 4 = 0. 2 to be factors. = 0. 1 + i) (x + 1 − i). And the four roots are 1, 2, 1 i, and 1+i. Ex. 2. To solve the equation x3- 5x + 2 = 0. The factors of x2+2x· · 1 are {x + (1 + √2)}{x + (1 − √2)}. A linear equation in any variable is simply a linear factor of unity. By proper transformations such an equation may always be brought to the form which is the only solution, a having but a single value. EXERCISE III. d. 1. Solve the following equations — i. x3 + 4x2 x 4 = 0. ii. x5+x3- x2 - 1 = 0. iii. x3+3x2 + 4x + 2 = 0. 2. Find values of x that will make (b2 − a2)x + equal CHAPTER IV. HIGHEST COMMON FACTOR. LEAST COMMON MULTIPLE. 61. The expressions 2 a2bc and 6ab2 have 2, a, and bas factors common to both, and the product of these, 2 ab, is the highest common factor of the expressions. The name Highest Common Factor is contracted to H. C. F., and sometimes G. C. M. (greatest common measure) is used in its stead. The expressions -7x+6 and 2+223-9x2+8 factor respectively into (-1) (x − 2) (x+3) and (x-1)(x+1)(x − 2) (x+4). They accordingly have the binomial factors (x-1) (x-2) or x2-3x+2 as their H. C. F. Common monomial factors, where they exist, are readily detected by inspection. To detect binomial factors, we may factor the expressions and pick out the common factors, as in the preceding example, or we may proceed upon the principle now to be established. 62. Theorem. If two expressions have a common factor, the sum and the difference of any multiples of the expressions have the same common factor. Let A and B denote the two expressions, and let f denote their common factor, so that A = Pf and B = Qf, where P and Q denote all the factors remaining in A and B respectively after the removal of f. Let a and b be any numerical multipliers. Then, aA±bB=aPƒ±bQf= (aP±bQ)ƒ; and as this last expression contains the common factor f, the theorem is proved. 63. Now let A and B be two integral functions of x, and let them have the common factor f, which we will suppose to be quadratic, as their H. C. F. By taking the sums or differences of proper mu.tip.es of A and B, we may reduce the dimensions of each by unity, and obtain two new functions A' and B', one dimension lower respectively than A and B, and of which ƒ is still a common factor. By operating in a similar manner upon A' and B', we find two functions A" and B", two dimensions lower respectively than A and B, and containing as a common factor. f By a continuation of this process we must eventually reduce A and B to depend upon functions of two dimensions, and having ƒ as a common factor. Hence these, upon rejecting all monomial factors, must be the factor f, and must therefore be identical. And thus the identity of the two results at any stage of the operation indicates that the H. C. F. is obtained. Ex. 1. Let A= 6 x3 — 7 x2 - 9 x 2, B=2x3 + 3x2 - 11x-6. - 9 x 2 B .2 x3+ 3x2 18 3 A. 16x2 - 24 x 16 2x2-3x 2 3 A-B 16 x3- 24 x2 - 16 x Reject factor 8x, 2 x2-3x-2 The results being identical shows that the highest common fac tor is 2x2 3x-2. |