(x-2)/(x^2+4x+3)x−2x2+4x+3
color(red) "Factorize the Denominator"Factorize the Denominator
x^2+4x+3 = (x+1)(x+3)x2+4x+3=(x+1)(x+3)
Our expression becomes
(x-2)/((x+1)(x+3))x−2(x+1)(x+3)
This expression can be written as
(x-2)/((x+1)(x+3)) = A/(x+1) + B/(x+2)x−2(x+1)(x+3)=Ax+1+Bx+2
(x-2)/((x+1)(x+3)) =(A(x+2)+B(x+1))/((x+1)(x+2))x−2(x+1)(x+3)=A(x+2)+B(x+1)(x+1)(x+2)
Equating the Numerators
x-2 = A(x+2)+B(x+1)x−2=A(x+2)+B(x+1)
Now we have to solve for AA and BB
Let us take x=-1x=−1 for making x+1 =0x+1=0 and thus removing BB.
-1-2 = A(-1+2)+B(-1+1)−1−2=A(−1+2)+B(−1+1)
-3=A −3=A
A=-3A=−3
Now let us use x=-2x=−2
-2-2=A(-2+2)+B(-2+1)−2−2=A(−2+2)+B(−2+1)
-4=-B−4=−B
B=4B=4
Substituting the value of AA and BB in
(x-2)/((x+1)(x+3)) = A/(x+1) + B/(x+2)x−2(x+1)(x+3)=Ax+1+Bx+2
Our final answer is
(x-2)/((x+1)(x+3)) = -3/(x+1) + 4/(x+2)x−2(x+1)(x+3)=−3x+1+4x+2
Rewriting
(x-2)/((x+1)(x+3)) = 4/(x+2)-3/(x+1) x−2(x+1)(x+3)=4x+2−3x+1
