SOLUTION: a) f(x)= x^2-1/x(x=2) find the partial fraction of f(x) b) Solve f(x)>0, hence write the solution for f(x)<0 Solution of the question need show in line and also table as well.

Algebra ->  Inequalities -> SOLUTION: a) f(x)= x^2-1/x(x=2) find the partial fraction of f(x) b) Solve f(x)>0, hence write the solution for f(x)<0 Solution of the question need show in line and also table as well.      Log On


   

Question 621023: a) f(x)= x^2-1/x(x=2)
find the partial fraction of f(x)
b) Solve f(x)>0, hence write the solution for f(x)<0
Solution of the question need show in line and also table as well.
tq.
Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
f(x) = %28x%5E2-1%29%2F%28x%28x-2%29%29 

When the degree of the numerator is greater than or equal to the
degree of the denominator, the fraction must be divided out first
by long division:

 %28x%5E2-1%29%2F%28x%28x-2%29%29 = %28x%5E2-1%29%2F%28x%5E2-2x%29%29

                      1
x² - 2x + 0)x² - 0x - 1
            x² - 2x + 0
                 2x - 1

f(x) = 1 + %282x-1%29%2F%28x%5E2-2x%29 = 1 + %282x-1%29%2F%28x%28x-2%29%29

f(x) = 1 + %282x-1%29%2F%28x%28x-2%29%29 = 1 + A%2Fx + B%2F%28x-2%29

%282x-1%29%2F%28x%28x-2%29%29 = A%2Fx + B%2F%28x-2%29

2x-1 = A(x-2) + Bx

This must be true for all values of x so we choose x=2
so the first term on the right will be 0

2(2)-1 = A(2-2) + B(2)
 4 - 1 = A(0) + 2B
     3 = 0 + 2B
     3 = 2B
     3%2F2 = B

Now we choose x=0 so the second term on the right will be 0

2(0)-1 = A(0-2) + B(0)
   0-1 = A(-2) + 0
    -1 = -2A 
    %28-1%29%2F%28-2%29 = A
    1%2F2 = A

A%2Fx + B%2F%28x-2%29 = expr%28%281%2F2%29%2Fx%29 + %283%2F2%29%2F%28x-2%29 = expr%28%281%2F2%29%2Fx%29·2%2F2 + %283%2F2%29%2F%28x-2%29·2%2F2 = 1%2F%282x%29 + 3%2F%282%28x-2%29%29

Therefore f(x) = %28x%5E2-1%29%2F%28x%28x-2%29%29 = 1 + 1%2F%282x%29 + 3%2F%282%28x-2%29%29


Edwin