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

Algebra ->  Test -> 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      Log On


   



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) About Me  (Show Source):
You can put this solution on YOUR website!
You should have grouped the factors in the denominator

 f(x) = (x^2 + 7x - 6)/[(x-1)(x-2)(x+1)]

f%28x%29+=%28x%5E2+%2B+7x+-+6%5E%22%22%29%2F%28%28x-1%29%28x-2%29%28x%2B1%29%5E%22%22%29%22==%22A%2F%28x-1%29%2BB%2F%28x-2%29%2BC%2F%28x%2B1%29

Multiply through by (x-1)(x-2)(x+1)

f%28x%29+=x%5E2%2B7x-6%22==%22A%28x-2%29%28x%2B1%29%2BB%28x-1%29%28x%2B1%29%2BC%28x-1%29%28x-2%29

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

f%28x%29+=-expr%281%2F%28x-1%29%29-expr%282%2F%28x%2B1%29%29%2B4%2F%28x-2%29  

Those terms look similar to the sum of an infinite geometric series:

f%28x%29+=a%2F%281-r%29%22%22=%22%22a%2Bar%2Bar%5E2%2Bar%5E3%2B%22%22%2A%22%22%2A%22%22%2A%22%22%22%22=%22%22a%281%2Br%2Br%5E2%2Br%5E3%2B%22%22%2A%22%22%2A%22%22%2A%22%22%29

We can get the first term in that form:

-1%2F%28x-1%29=1%2F%28-x%2B1%29=1%2F%281-x%29

We can get the second term in that form:

-2%2F%28x%2B1%29=-2%2F%281%2Bx%29=-2%2F%281-%28-x%29%29

The third term needs to have first term 1 in the denominator:

4%2F%28x-2%29=-4%2F%282-x%29=-4%2F%282%281-%22x%2F2%22%29%29=-2%2F%281-%22x%2F2%22%29

So we have:

f%28x%29+=1%2F%281-x%29-2%2F%281-%28-x%29%29-2%2F%281-%22x%2F2%22%29

Using the above equation for a%2F%281-r%29 on each term:



Simplifying that gives









Simplifying and omitting terms in x4 and higher we have:

f%28x%29+=-3+%2B+2x+-+expr%283%2F2%29+x%5E2+%2B+expr%2811%2F4%29x%5E3

Edwin