Express (3x^2 - 3x - 2)/(x-1)(x-2) in partial fractions

Step 1:

When asked to exrpess something in partial fractions, we first compare the power of the numerator to the power of the denominator.

In our case we have that the power of quadratic equation in the numerator is equal to 2, while the power of the denominator is 

(x-1)(x-2) = x- 3x +2 

which is equal to 2 as well.

Step 2:

Now we devide the numerator by the denominator.

Using long division we get that

(3x2 - 3x - 2)/(x-1)(x-2) = 3 + (6x-8)/(x- 3x + 2)

Step 3:

The next step is express (6x-8)/(x- 3x + 2) as a partial fraction

(6x-8)/(x- 3x + 2) = (6x-8)/(x-1)(x-2) 

(6x-8)/(x-1)(x-2) = A/x-1 + B/x-2

Step 4:

Now we multiply both the LHS and the RHS by (x-1)(x-2) because this leads to a common denominator.

6x - 8 = A(x-2) + B(x-1)

Now we have to use two different values for x, such that in the first instance B=0, and in the second instance,  A=0

Hence, when x=1,

6(1) - 8 = A(1-2) + B(1-1)

6 - 8 = -A + 0

-2 = - A

A = 2

When x=2

6(2) - 8 = A(2-2) + B(2-1)

12 - 8 = 0 + B

B = 4

Step 5:

Now going back to our original equation,

(3x2 - 3x - 2)/(x-1)(x-2) = 3 + (6x-8)/(x- 3x + 2)

 3 + (6x-8)/(x- 3x + 2) = 3 +  A/x-1 + B/x-2

and using A=2 , B=4 we get

3 + (6x-8)/(x- 3x + 2) = 3 +  2/x-1 + 4/x-2

Hence, our desired result is

(3x2 - 3x - 2)/(x-1)(x-2) =  3 +  2/x-1 + 4/x-2

PK
Answered by Pantelis K. Maths tutor

18101 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Solve the equation " 2sec^2(x) = 5tanx " for 0 < x < π


Differentiate with respect to x and write in its simpliest form, Y=(2x-3)/x^2?


Express cos(2x) in terms of acos^2(x) + b


if f is defined on with f(x)=x^2-2x-24(x)^0.5 for x>=0 a) find 1st derivative of f, b) find second derivative of f, c) Verify that function f has a stationary point when x = 4 (c) Determine the type stationary point.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences