CHAPTER X. RESOLUTION INTO FACTORS. 86. DEFINITION. When an algebraic expression is the product of two or more expressions, each of these latter quantities is called a factor of it, and the determination of these quantities is called the resolution of the expression into its factors. 87. Rational expressions do not contain square or other roots (Art. 14) in any term. 88. Integral expressions do not contain a letter in the denominator of any term. Thus, x2+3 xy + 2 y2, and 1 x2 + 1⁄2 xy — — y2 are integral expressions. 89. In this chapter we shall explain the principal rules by which the resolution of rational and integral expressions into their component factors, which are rational and integral expressions, may be effected. WHEN EACH OF THE TERMS IS DIVISIBLE BY A COMMON FACTOR. 90. The expression may be simplified by dividing each term separately by this factor, and enclosing the quotient within brackets; the common factor being placed outside as a coefficient. Ex. 1. The terms of the expression 3 a2 - 6 ab have a common factor 3 a. ... 3 a2 6 ab = 3 a(a - 2 b). Ex. 2. 5 a2bx3 15 abx2 - 20 b3x2 = 5 bx2 (a2x — 3 a − 4 b2). WHEN THE TERMS CAN BE GROUPED SO AS TO 91. Ex. 1. Resolve into factors x2 · ax + bx - ab. Noticing that the first two terms contain a factor x, and the last two terms a factor b, we enclose the first two terms in one bracket, and the last two in another. Thus, since each bracket of (1) contains the same factor x — a. Ex. 2. Resolve into factors 6 x2 - 9 ax + 4 bx - 6 ab. = (12 a2 - 4 ab) — (3 ax2 - bx2) · (3 a − b) (4 a − x2). NOTE. In the first line of work it is usually sufficient to see that Any suitably chosen pairs Thus, in the last example, by a each pair contains some common factor. 12 a2 - 4 ab - 3 ax2 + bx2 = = (12 a2 - 3 ax2)-(4 ab - bx2) 3 a(4 a x2)b(4 α- x2) = (4a-x) (3 a - b). The same result as before, for the order in which the factors of a product are written is of course immaterial. 92. When the Coefficient of the Highest Power is Unity. Before proceeding to the next case of resolution into factors the student is advised to refer to Chap. IV. Art. 51. Attention has there been drawn to the way in which, in forming the product of two binomials, the coefficients of the different terms combine so as to give a trinomial result. Thus, by Art. 51, We now propose to consider the converse problem: namely, the resolution of a trinomial expression, similar to those which occur on the right-hand side of the above identities, into its component binomial factors. By examining the above results, we notice that: 1. The first term of both the factors is x. 2. The product of the second terms of the two factors is equal to the third term of the trinomial; thus in (2) above we see that 15 is the product of 5 and -3; while in (3) 15 is the product of +5 and 3. 3. The algebraic sum of the second terms of the two factors is equal to the coefficient of x in the trinomial; thus in (4) the sum of − 5 and +3 gives — 2, the coefficient of x in the trinomial. In showing the application of these laws we will first consider a case where the third term of the trinomial is positive. Ex. 1. Resolve into factors x2 + 11x + 24. The second terms of the factors must be such that their product is +24, and their sum + 11. It is clear that they must be + 8 and + 3. ... x2 + 11 x + 24 = (x+8)(x+3). Ex. 2. Resolve into factors x2 - 10x + 24. The second terms of the factors must be such that their product is +24, and their sum 10. Hence they must both be negative, and it Ex. 5. Resolve into factors x2. The second terms of the factors +10 a2, and their sum - 11 ax + 10 a2. must be such that their product is - 11 a. Hence they must be — 10 a and •. x2 — 11 ax + 10 a2 = (x − 10 a) (x a). a. NOTE. In examples of this kind the student should always verify his results, by forming the product (mentally, as explained in Chapter IV.) of the factors he has chosen. 27. x2 + 49 xy + 600 y2. 28. x2y2 + 34 xy + 289. 29. a4b4+37 a2b2 + 300. 30. a2 - 29 ab + 54 b2. 31. x4162x2 + 6561. 32. 12-7x+ x2. 33. 20+ 9x + x2. 35. 88+ 19 x + x2. 37. 204-29 x2 + x1. 38. 21635x + x2. 93. Next consider a case where the third term of the trinomial is negative. The second terms of the factors must be such that their product is – 35, and their algebraic sum + 2. Hence they must have opposite signs, and the greater of them must be positive in order to give its sign to their sum. The required terms are therefore +7 and - 5. .. x2+2x-35 = (x + 7) (x − 5). Ex. 2. Resolve into factors x2 - 3 x — 54. The second terms of the factors must be such that their product is - 54, and their algebraic sum - 3. Hence they must have opposite signs, and the greater of them must be negative in order to give its sign to their sum. The required terms are therefore 9 and +6. .. x2 - 3 x 54 = (x − 9) (x + 6). Remembering that in these cases the numerical quantities must have opposite signs, if preferred, the following method may be adopted. Ex. 3. Resolve into factors x2y2 + 23 xy - 420. Find two numbers whose product is 420, and whose difference is 23. These are 35 and 12; hence inserting the signs so that the positive may predominate, we have x2y2+23 xy - 420 = (xy + 35) (xy — 12). |