Final answer:
To find the partial fraction decomposition of the given rational expression (10x/(2x² - 3x)), the denominator (2x² - 3x) is factored as x(2x - 3) and the rational expression is written in the form of partial fractions. Finally, the partial fraction decomposition is obtained as 5/x - 3/(2x - 3).
Step-by-step explanation:
To find the partial fraction decomposition of the given rational expression, we first factor the denominator.
The denominator, 2x² - 3x, can be factored as x(2x - 3).
Next, we write the rational expression in the form of partial fractions:
10x/(2x² - 3x) = A/x + B/(2x - 3)
To find the values of A and B, we multiply through by the common denominator, (2x - 3)x:
10x = A(2x - 3) + B(x)
Expanding this equation, we have:
10x = (2A + B)x - 3A
From this equation, we can see that A = 5 and B = -3.
Therefore, the partial fraction decomposition of the given rational expression is:
10x/(2x² - 3x) = 5/x - 3/(2x - 3)