242 



Involution of Polynomials. 



Ex. 3. Find the coefficients of a-{-h-\-c-\-d-\-e 



ahcde 



a'^bcd 



a^bc 



a^b^ 



a'b 



Hence a-\-b-{-c+d+e^ =a^-]-oa*b-{-10a''b^-\-2Qa = bc-\^30a''b^c 

 '\-^0a^bcd-i-120abcde-\-^c. 



If \he number of terms in the root is greater than the index of 

 the powet, the excess produces no change in the coefficients, as 

 no more than n letters can enter any term of the power. 



Ex. 4. Find the coefficients of a + b-\-c+d\ 



The binomial coefficients are 1, 6, 15, 20. The introduction 

 of the third term c, gwes us 2A=30, 3B=60. And the fourth 

 term d, gives 2-3B = 120, ^-40 = 180. 



Hence a-\-b-^c+d^ =a' +^a'b-{-15a'b^ +2(}a^b^ + 30a*5c 

 -}-e>0a^b''C-{-120a'bcd+lS0a'b^cd^&cc. 



Or we might have found P as before, observing that as two 

 terms are deficient in the root, P must have at least two divisions 

 before the consequent coefficient can enter into the expression of 

 the power. 



P=6x5x4x3x2=720 



P_ 



2" 



2^~ " 



^ 



2-3- ■ 

 P 



- 360 



180 



120 



2-2-3 



60 



abcdef 

 a^bcde 



a^b'^cd 



a^bcd 



a^b^c 



