Write the ratio (3x+2)/(x^2+x) using partial fraction decomposition.
We have to write (3x+2)/(x^2+x) as sum of fractions.
Now (x^2 + x )= x ( x +1)
=> A / (x+1) + B/x
=> (Ax + Bx + B)/ x(x+1)
So we get A + B = 3 and B = 2
As B = 2 , A = 3 - 2 = 1
So the required form is 1/(x+1) + 2/x.
Therefore (3x+2)/(x^2+x) = 1/(x+1) + 2/x.
To decompose a fraction we'll have to factor the denominator. We'll factorize by x:
x^2+x = x( x + 1 )
We'll write the fractions with one of the factors for each of the denominators.
(3x+2)/(x^2+x) = A/x + B/( x + 1 )
We'll multiply both side by the common denominator x( x + 1 ):
3x+2 = A(x+1) + Bx
We'll remove the brackets:
3x+2 = Ax + A + Bx
We'll factorize by x to the right side:
3x+2 = x(A+B) + A
We'll compare and we'll get:
A+B = 3
A = 2
2 + B = 3
B = 3 - 2
B = 1
So, the fraction (3x+2)/(x^2+x) = 2/x + 1/(x + 1)