Unknown

108 3. Factors and Zeros

in the form

-- A B

t-1 +t+4

for suitable constants A and B.

4. (4 Let N/q(t) b e a rational function for which the degree of the
   numerator is less than that of the denominator and the de-
   nominator q(t) is the product of h distinct linear factors t - si
   (1 5 i 5 k). Show that p(t)/q(t) can be written in the form

,g, -- (t TQi)

for suitable constants ci.

(b) Prove that, in (a), the ci are uniquely determined by the formula

Ci = P(Q)/d(Qi).

(c) Deduce from (a) a representation of that polynomial f(t) of

degree less than Jz for which f(si) = bi, where, for 1 5 i 5 k, bi

are assigned values.

5. In each case, use a partial fraction representation of ak to determine
   C{ak : lc=2,3,... , n}, where ak is equal to

(4 l/W - 111

(b) l/P3 - 4

(c) (kt + l)((kl- 1)X + 1)’

6. The partial fraction decomposition can be extended to the situation
   in which q(t) is a product of irreducible factors some of whose degrees
   are greater than 1. In this case, for each such factor v(t), there is a
   summand of the form u(t)/v(t) where deg u(t) < deg v(t).

(4

(b)

Verify that t4 - 108t + 243 = (t - 3)2(t2 + 6t + 27).

The rational function

t3 + t2 + 15t - 27

t4 - 108t + 243

can be written in the

form

--

t _” 3 + (t :3)2 + t2 C+t6;:27 ’

Determine the constants A, B, C, D and check your answer.

7. Express the rational function
   rw = 7t2 - 2t + 3
   24 - 3t3 + t2 - 3
   as a sum of partial fractions, one associated with each real irreducible
   factor of the denominator.