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Answers to Exercises; Chapter (^3 283)
1 1 1 n-l
=-
[
--- =
x x+1 7x2 +^1 1 (x+ l)(nx+ 1)’
6.6. We have t3+t2+15t-27 = A(t-3)+B+(Ct+D)(t-3)2.
Setting t = 3 yields B = 1. Comparison of coefficients leads to A = 2/3,
B = 1, C = l/3, D = 0.
6.7. t4 - 3t3 + t2 - 3t = t(t - 3)(t2 + 1). The rational function has the
representation
1 2
t+
-t + 1
-- -+-.
t-3 t2 + 1
6.8.
111
[
1
a=-^2 --- t-1 t+1 I
1
1
[
(^1) t+2
m=-

3 t-1 t2+t+1 1
1 1
= 3 [- t-1+t-w A+$$2 1
where w = (-1 + i&)/2.
1 1 1 1 2

• t4 -^1 = ------^4 [ t-1 t+1 t2 +^1 1
  1 1 1. i
  = 4 [ --- t-1 t+1 +2-- t-i t+i 1

1 1

[

(^1) t3 + 2t2 + 3t + 4

• =

  ---

  
  

  t5- 1 5 t-1 t4+@+P+t+1 I
  1 1
  [

  
  

---- c c2 c3 c4

= 5 t-1+&C+ t--2+&(3+-. t - p 1

If (t” - l)-’ = ~~~~ ui(t - <A)-‘, then ai must satisfy

uj fl(c; - <i) = 1 or Ui<~n~(Iv<~) - = 1.

j#i i=l

Since the product is equal to t”-’ +... + t + 1 evaluated at t = 1, we find
that ai = CL/n.