SOLUTION: Let f(x) = (x^2 + 7x - 6)/(x-1)(x-2)(x+1)
(i) Express f(x) in partial fractions.
(ii) Show that, when x is sufficiently small for x^4 and higher powers to be neglected, f(x) = -3
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-> SOLUTION: Let f(x) = (x^2 + 7x - 6)/(x-1)(x-2)(x+1)
(i) Express f(x) in partial fractions.
(ii) Show that, when x is sufficiently small for x^4 and higher powers to be neglected, f(x) = -3
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Question 1205608: Let f(x) = (x^2 + 7x - 6)/(x-1)(x-2)(x+1)
(i) Express f(x) in partial fractions.
(ii) Show that, when x is sufficiently small for x^4 and higher powers to be neglected, f(x) = -3 + 2x - (3/2 x^2) + (11/4 x^3) Answer by Edwin McCravy(20054) (Show Source):
You should have grouped the factors in the denominator
f(x) = (x^2 + 7x - 6)/[(x-1)(x-2)(x+1)]
Multiply through by (x-1)(x-2)(x+1)
Substituting x=2 will cause two of the terms on the right to become 0,
then you can solve for B = -2
Substituting x=-1 will also cause two of the terms on the right to become 0,
then you can solve for C = 4
Substituting x=1 will also cause two of the terms on the right to become 0,
then you can solve for A = -1
Those terms look similar to the sum of an infinite geometric series:
We can get the first term in that form:
We can get the second term in that form:
The third term needs to have first term 1 in the denominator:
So we have:
Using the above equation for on each term:
Simplifying that gives
Simplifying and omitting terms in x4 and higher we have:
Edwin