to factor 4 a2—9 b2+4 a−6 b, we can of course factor the part 4 a2-9 b2, giving (2 a+3 b) (2 a−3 b), but this does not give us the factors of the given (whole) expression, 4 a2-9 b2+4 a-6 b. Observe that any expression may be said to have been factored only when it has all been put into the form of a product of two or more factors. 78. Summary of Factoring. All the examples in factoring which we have thus far considered have been worked by use of the following formulas : Each expression in the following list may be factored by the formulas in § 78. Either a single one of the formulas is necessary, or several of them in the manner shown in § 77. Before attempting these, the pupil will find it desirable to review the exercises in factoring in Chapter VI and read § 77 carefully. Factor each of the following expressions. 1. x2-ax+cx-ac. [HINT. Write as x(x−a)+c(x—a).] 2. y3+y2+y+1. [HINT. Write as y2(y+1)+y+1.] [HINT. Write as (23-22)-(2-1).] K 4. 2 x3-8 x2y+8 xy2. [HINT. Write as 2 x(x2-4 xy+4 y2) and apply Formnla VI, § 78.] 5. x2-11x+30. [HINT. This comes under Formula IV, § 78. See § 58.] 28. xy-1+x-y. 29. a2+(b-2 bx2)ay-2 b2x2y2. 30. 1-a2b2x2y2+2 abxy. 31. (a+b)2(x−y) − (a+b)(x2 — y2). 32. (x2 y2)2- (x2 — xy)2. 33. m3+n2-mn-mn2. *36. 5 x1y-5 xya. 38. (1-2x)2-x4. 39. x3y-10 x2y2z2+25 xy3z1. 40. x4-18 x2+81. *41. 1+(x+1)3. *42. x3+15 x2+75x+125. *43. 3 ab(a+b)+a3+b3. Famous in mathematics for his researches, which were of a most advanced kind, and especially famous in astronomy for his enunciation of the Nebular Hypothesis. Interested also in physics and at various times held high political offices under Napoleon. CHAPTER VIII HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE PART I. HIGHEST COMMON FACTOR 79. Common Factors. In arithmetic a factor of each of two or more numbers is called a common factor of the numbers. Thus, 3 is a common factor of 9 and 15; 5 is a common factor of 10, 15, and 25. In the same way, we say in algebra that x is a common factor of 2 x and 5 x; У is a common factor of 3 y and y2; rs is a common factor of 2 r2s2 and rs; a+b is a common factor of (a+b)2 and a(a+b). 80. Prime Factors. A number that has no factor except itself and unity is called in arithmetic a prime number. Such a number when used as a factor is called a prime factor. Thus, the prime factors of 10 are 5 and 2; the prime factors of 36 are 3, 3, 2, and 2. In the same way, we say in algebra that the prime factors of a2b(a2 — b2) are a, a, b, a−b, and a+b. 81. To Find Common Factors. As soon as we factor each of several numbers into their prime factors, we can easily pick |