Question

Express 2x+1/(x-2)(x²+1) as a partial fraction

Answer 1

2x+1/(x-2)(x²+1)= 1/(x-2) + (-2x )/(x²+1)

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)