Question

Resolve
x^3/(x-1)(x-2)
into partial fractions​

Answer 1

Answer:

x + 3 - 1/(x-1) + 8/(x-2)

Step-by-step explanation:

First, carrying out the division we find

     x³ / ( ( x - 1 ) ( x - 2 ) )

 =  x³ / ( x² - 3x + 2 )

 =  x + 3 + ( 7x - 6 ) / ( ( x - 1 ) ( x - 2 ) )

Now we need to write the remainder as partial fractions.  We seek A and B such that

( 7x - 6 ) / ( ( x - 1 ) ( x - 2 ) )  =  A / ( x - 1 )  +  B / ( x - 2 )

Multiplying through by (x-1)(x-2), this becomes

7x - 6 = A ( x - 2 ) + B ( x - 1 )

Putting x = 1, we get

1 = -A  ⇒  A = -1

Putting x = 2, we get

8 = B

So the answer is

     x³ / ( ( x - 1 ) ( x - 2 ) )  =  x + 3 - 1/(x-1) + 8/(x-2)

Hope this helps!