V. ON THE RESOLUTION OF EXPRESSIONS INTO FACTORS. 86. WE shall discuss in this Chapter an operation which is the opposite of that which we call Multiplication. In Multiplication we determine the product of two given factors: in the operation of which we have now to treat the product is given and the factors have to be found. 87. For the resolution, as it is called, of a product into its component factors no rule can be given which shall be applicable to all cases, but it is not difficult to explain the process in certain simple cases. We shall take these cases separately. 88. CASE I. The simplest case for resolution is that in which all the terms of an expression have one common factor. This factor can be seen by inspection in most cases, and therefore the other factor may be at once determined. 89. CASE II. The next case in point of simplicity is that in which four terms can be so arranged, that the first two have a common factor and the last two have a common factor. 90. Before reading the Articles that follow the student is advised to turn back to Art. 56, and to observe the manner in which the operation of multiplying a binomial by a binomial produces a trinomial in the Examples there given. He will then be prepared to expect that in certain cases a trinomial can be resolved into two binomial factors, examples of which we shall now give. 91. CASE III. To find the factors of x2+7x+12. Our object is to find two numbers whose product is 12, and whose sum is 7. These will evidently be 4 and 3, :.x2+7x+12=(x+4)(x+3). Again, to find the factors of x2+5bx+6b2. Our object is to find two numbers whose product is 662, These will clearly be 36 and 26, and whose sum is 5b. •. x2+5bx+6b2 = (x + 3b) (x+2b). 8. x2+13mx + 36m2. 16. n2+27nq+140g2. 92. CASE IV. To find the factors of x2-9x+20. Our object is to find two negative terms whose product is 20, and whose sum is -9. These will clearly be -5 and -4, Our object is to find two terms, one positive and one negative, whose product is -84, and whose sum is 5. These are clearly 12 and -7, • +50-84= ( + 12)(7). Our object is to find two terms, one positive and one negative, whose product is -28, and whose sum is -3. These will clearly be 4 and -7, .. x2-3x-28=(x+4)(x-7). 95. The results of the four preceding articles may be thus stated in general terms: a trinomial of one of the forms x2+ax+b, x2—ax+b, x2+ax−b, x2 — ax−b, may be resolved into two simple factors, when b can be resolved into two factors, such that their sum, in the first two forms, or their difference, in the last two forms, is equal to a. 96. We shall now give a set of Miscellaneous Examples on the resolution into factors of expressions which come under one or other of the cases already explained. 97. We have said, Art. 45, that when a number is multiplied by itself the result is called the Square of the number, and that the figure 2 placed over a number on the right hand indicates that the number is multiplied by itself. Thus a2 is called the square of a, (x-y) is called the square of x-y. The SQUARE ROOT of a given number is that number whose square is equal to the given number. Thus the square root of 49 is 7, because the square of 7 is 49. So also the square root of a2 is a, because the square of a is a2 and the square root of (x-y)2 is x-y, because the square of x-y is (x-y)2. The symbol placed before a number denotes that the square root of that number is to be taken: thus 25 is read "the square root of 25." Note. The square root of a positive quantity may be either positive or negative. For since a multiplied by a gives as a result a2, and - a multiplied by a gives as a result a2, it follows, from our definition of a Square Root, that either a or a may be regarded as the square root of a2. But throughout this chapter we shall take only the positive value of the square root. |