Expressx(1 − 2x)2(1 − 3x)in partial fractions.

To express the given function in partial fractions, we first need to factorize the denominator. However, in this case, the denominator is already factorized as (1 - 2x) and (1 - 3x).

The general form of the partial fraction decomposition for this function is:

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

where A and B are constants that we need to find.

To find A and B, we can use the method of equating coefficients.

First, we clear the fractions by multiplying both sides by the common denominator (1 - 2x)(1 - 3x):

x = A(1 - 3x) + B(1 - 2x)

Now, we can equate the coefficients on both sides of the equation to find A and B.

For the x term:

0 = -3A - 2B

For the constant term:

1 = A + B

Solving these two equations simultaneously, we get A = 2 and B = -1.

So, the partial fraction decomposition of the given function is:

x/(1 - 2x)(1 - 3x) = 2/(1 - 2x) - 1/(1 - 3x)