2 PRACTICAL MATHEMATICS 



2. The previous method can be replaced by the solution of 

 a quadratic equation ; for if * = a : makes the expression 



b c 



x 2 -\ x + - disappear, then x <x., is a factor. Also if x = a, 

 a a 



makes the expression disappear, then x <x 2 is a factor. Con- 

 sequently a(x a 1 )( a 2 ) will be the factors of ax 2 + bx + c 

 where a x and a 2 are the roots of the quadratic equation 



x * + \u + = o. 



a a 

 To find the factors of 5x 2 - 7x - 22 



5x 2 - 7x - 22 = 5(x 2 - l-4x - 4-4) 



Solving the equation x 2 l-4# 4-4 =0 



x 2 - 1-40 + (0-7) 2 = 4-89 



x - 0-7 = 2-211 



x = 2-911 or - 1-511 



The factors are 5(x - 2-911) (x + 1-511) 



3. Partial Fractions. For the integration of algebraic fractions 

 it is necessary that the fraction must be expressed in its simplest 

 and most convenient form for integration. For such purposes a 

 fraction is much better dealt with when it is expressed as the 

 sum or difference of simpler fractions. These simpler fractions 

 are spoken of as " Partial Fractions," and the number of partial 

 fractions which can represent a given fraction depends upon the 

 number of factors, linear or otherwise, in the denominator of 

 that fraction. 



If, for example, the denominator contains three factors, 

 then there will be three partial fractions, the respective de- 

 nominators of which are the three factors taken in order. Thus, 



3x + 2 ABC 



can be written as H + 



(x - 2)(x + 3)(2x - 5) X - 2 ' x + 3 ' 2x - 5 



providing the necessary values of A, B, and C are found. Also 



x 2 x 2 



, , w - rr or , 777 .,.. , rr can be written as 



(x + l)(x s I) (x + l)(x l)(x 2 + x + 1) 



A B Cx +D 



Care must be taken that the numerator of any partial fraction 

 shall always be of one degree less than that of its denominator. 



4. Our work in partial fractions can be divided up into four 

 different cases. 



Case I. When the denominator of the fraction is the product 

 of a certain number of different linear factors. 



3x + 2 A . B C 



(x - 2)(x + 3)(2x -5) x-2x+32x-5 



