Answer:
2x+1/(x-2)(x²+1)= 1/(x-2) + (-2x )/(x²+1)
Step-by-step explanation:
2x+1/(x-2)(x²+1) = a/(x-2) + (bx+c)/(x²+1)
Multiplying with the denominators
2x+1= a(x²+1) +( bx+c)(x-2)
Let x = 2
2(2) + 1 = a(2² + 1) + 0
5 = a5
a = 1
Then let's expand the brackets
2x +1= ax² + a + bx² -2bx + cx - 2c
Comparing co-efficients
ax² + bx² = 0
a + b = 0 equation 1
a -2c = 1 equation 2
-2bx + cx = 2x
-2b + c = 2 equation 3
Let's remember a= 1
a + b = 0
1 + b= 0
b = -1
a -2c = 1
1 - 2c = 1
c= 0
-2b + c = 2
-2(-1) + 0 = 2
2= 2. Verified.
a/(x-2) + (bx+c)/(x²+1)
= 1/(x-2) + (-2x )/(x²+1)
2x+1/(x-2)(x²+1)= 1/(x-2) + (-2x )/(x²+1)
