To rewrite the given function using partial fractions, we need to express the rational function in a form where it is broken down into simpler fractions. The function given is:

$f(x) = \frac{4x - 3}{x^2(x - 2)}$

Step 1: Set up the partial fraction decomposition.

We will decompose the expression into terms based on the factors in the denominator: $x^2(x - 2)$. The general form of the partial fraction decomposition is:

$\frac{4x - 3}{x^2(x - 2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2}$

Here, $A$, $B$, and $C$ are constants we need to determine.

Step 2: Combine the right-hand side over a common denominator.

To do that, express the terms on the right-hand side with the common denominator $x^2(x - 2)$:

$\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2} = \frac{A(x)(x - 2) + B(x - 2) + C(x^2)}{x^2(x - 2)}$

Now we can simplify the numerator on the right-hand side:

$A(x)(x - 2) = A(x^2 - 2x)$ $B(x - 2) = Bx - 2B$ $C(x^2) = Cx^2$

So the expression becomes:

$A(x^2 - 2x) + B(x - 2) + Cx^2 = (A + C)x^2 + (B - 2A)x - 2B$

Step 3: Equate the numerators.

Now, equate the numerator of this expression to the numerator of the original fraction $4x - 3$:

$(A + C)x^2 + (B - 2A)x - 2B = 4x - 3$

Step 4: Solve for $A$, $B$, and $C$.

By comparing the coefficients of powers of $x$, we get the following system of equations:

1. $A + C = 0$ (coefficient of $x^2$),
2. $B - 2A = 4$ (coefficient of $x$),
3. $-2B = -3$ (constant term).

From the third equation, we can solve for $B$:

$-2B = -3 \quad \Rightarrow \quad B = \frac{3}{2}$

Now substitute $B = \frac{3}{2}$ into the second equation:

$\frac{3}{2} - 2A = 4 \quad \Rightarrow \quad -2A = 4 - \frac{3}{2} = \frac{8}{2} - \frac{3}{2} = \frac{5}{2}$

$A = -\frac{5}{4}$

Finally, use $A + C = 0$ to find $C$:

$-\frac{5}{4} + C = 0 \quad \Rightarrow \quad C = \frac{5}{4}$

Step 5: Write the final partial fraction decomposition.

Substitute the values of $A$, $B$, and $C$ into the partial fraction decomposition:

$\frac{4x - 3}{x^2(x - 2)} = \frac{-\frac{5}{4}}{x} + \frac{\frac{3}{2}}{x^2} + \frac{\frac{5}{4}}{x - 2}$

This is the partial fraction decomposition of the given function.

Would you like any further details or explanations?

Relative Questions:

1. How would the decomposition change if the denominator had complex roots?
2. Can we apply the same method for higher-order factors in the denominator, such as $x^3$?
3. What are the conditions under which partial fraction decomposition can be applied?
4. Can partial fractions be used in integral calculus? If so, how?
5. How would the procedure change if the numerator was of higher degree than the denominator?

Tip:

When performing partial fraction decomposition, always check if the degree of the numerator is lower than the degree of the denominator. If not, perform polynomial long division first.