in which Q(s)and R(s)are the quotient and remainder polynomials, respectively, with the division made so that the degree of R(s)is less than or equal to that of A(s) II. Factorization of Denominator PolynomiaL. The next step of the partial fraction expansion method entails e factorizing of the nth-order denominator polynomial A(s) into a product of n first-order factors. This factorization is always possible and results in the equivalent representation of A(s) as given by A(s)=(s-p1)(s-p2)…(s-pn) The terms P1, P2,...,Pn constituting this factorization are called the roots of polynomial A(s), or the poles of X(s) Ill. Partial Fraction Expansion. With this factorization of the denominator polynomial accom rational Laplace transform X(s) can be expressed as B(s) bs"+bn. (6.2) A(s)(s-P1)(s-P2)-(s-Pn) We shall now equivalently represent this transform function as a linear combination of elementary transform Case 1: A(s) Has Distinct Root X(s) PI s- p3 pn where the ag are constants that identify the expansion and must be properly chosen for a valid representation (s-Pr)X(ss-p, for k= 1, The expression for parameter ao is obtained by letting s become unbounded (i. e, s=+oo)in expansion(6.2). Case 2: A(s) Has Multiple Roots. X(S) B(s) B(s A(s)(s-p1)A1(s) The appropriate partial fraction expansion of this rational function is then given by (s-P1) other elementary terms due to the ts of© 2000 by CRC Press LLC in which Q(s) and R(s) are the quotient and remainder polynomials, respectively, with the division made so that the degree of R(s) is less than or equal to that of A(s). II. Factorization of Denominator Polynomial. The next step of the partial fraction expansion method entails the factorizing of the nth-order denominator polynomial A(s) into a product of n first-order factors. This factorization is always possible and results in the equivalent representation of A(s) as given by The terms p1, p2, . . ., pn constituting this factorization are called the roots of polynomial A(s), or the poles of X(s). III. Partial Fraction Expansion. With this factorization of the denominator polynomial accomplished, the rational Laplace transform X(s) can be expressed as (6.2) We shall now equivalently represent this transform function as a linear combination of elementary transform functions. Case 1: A(s) Has Distinct Roots. where the ak are constants that identify the expansion and must be properly chosen for a valid representation. and a0 = bn The expression for parameter a0 is obtained by letting s become unbounded (i.e., s = +•) in expansion (6.2). Case 2: A(s) Has Multiple Roots. The appropriate partial fraction expansion of this rational function is then given by As s p s p s pn ( ) ( )( ) . . . ( ) =- - - 1 2 X s B s A s bs b s b s ps p s p n n n n n ( ) ( ) ( ) ( )( ) ( ) = = + + ×××+ - - ××× - - - 1 1 0 1 2 X s sp sp sp n n ( ) = + - + - + ×××+ - a aa a 0 1 1 2 2 ak k s p Xs k n s pk =- = = ( ) ( ) , ,..., for 1 2 X s B s A s B s s p As q ( ) ( ) ( ) ( ) ( ) () = = - 1 1 X s sp sp n q A s q q ( ) () () = + ( ) - + ×××+ - + - ( ) a a a 0 1 1 1 1 other elementary terms due to the roots of 1