Problem 62 - Precalculus with Limits

Question

For the rational expression \(\frac{2 x+3}{x^{2}(x+2)^{2}},\) the partial fraction decomposition is of the form \(\frac{A x+B}{x^{2}}+\frac{C x+D}{(x+2)^{2}}\)

Step-by-Step Solution

Answer

After solving the system of equations, the values of A, B, C, and D will be found. Substitute these values back into the partial fractions decomposition, and the final solution will be the expression in the correct form.

Step 1: Write the General Form

In the partial fractions decomposition, write the general form as following: \( \frac{2 x+3}{x^{2}(x+2)^{2}} = \frac{A x+B}{x^{2}} + \frac{C x+D}{(x+2)^{2}} \)

Step 2: Multiply Through by the Common Denominator

Next, multiply every term by the common denominator of the right side to eliminate the fractions. The equation will become: \( 2x + 3 = (A x + B)(x + 2)^{2} + (C x + D) x^{2} \)

Step 3: Distribute and Combine Like Terms

Expand the right side of the equation and combine like terms. Now, gather terms and organize by descending powers of x.

Step 4: Equate Coefficients

Now that both sides of the equation are written in the same form, equate coefficients for the powers of x on both sides. This will give us a system of equations.

Step 5: Solve the System of Equations

Now solve the system of equations for the variables A, B, C and D. The solutions are the coefficients we need for the partial fraction decompositions.

Key Concepts

Rational Expressions

A rational expression resembles a fraction where the numerator and denominator are polynomials. In our exercise, the original expression is \( \frac{2x+3}{x^2(x+2)^2} \). This expression requires you to understand how both the numerator and denominator function together, indicating how the entire fraction behaves.
When dealing with rational expressions, be on the lookout for the degree of the polynomials. The degree of the polynomial in the denominator often determines the number of terms in a partial fraction decomposition. Here, the denominator \( x^2(x+2)^2 \) has already been factored, showing distinct linear and repeated quadratic factors.
The key task in partial fraction decomposition is breaking down this rational expression into simpler fractions that are easier to integrate or manipulate. The expression can be rewritten as a sum of fractions, each with polynomial numerators over the original denominator's factors.- In the exercise, partial fractions were split into: \( \frac{Ax+B}{x^2} \) and \( \frac{Cx+D}{(x+2)^2} \)- This decomposition gives clarity on how each part of the original fraction contributes to the total, simplifying complex processes like integration.

Coefficients

Coefficients are basically numbers or constants placed in front of variables like \( x \) within polynomial expressions. They play a crucial role in defining the size, direction, and proportion of each variable term in an equation.
In the partial fraction decomposition exercise, coefficients shine when we rewrite a complex fraction into smaller parts and equate like terms. By expanding and simplifying, you gain equations that are based on these coefficients.
Your goal here is to find the values of \( A, B, C, \) and \( D \) which are the coefficients of the decomposed fractions. They appear in expressions like \( (Ax + B)(x + 2)^2 \) and \( (Cx + D)x^2 \). To find these values:- Expand each part across the equation.- Match the coefficients of the same powers of \( x \) between both the sides of the equation.
This ultimately forms a system of equations that can be solved, giving the necessary coefficients to completely understand your decomposed partial fractions.

System of Equations

The system of equations comes into play during the process of equating coefficients from the expanded expression of our partial fraction decomposition. This involves setting the coefficients of terms with the same powers of \( x \) equal to each other on both sides of the equation.
For instance, expanding the partial fraction terms results in an equation like \( 2x + 3 = (Ax + B)(x + 2)^2 + (Cx + D)x^2 \). After expansion and simplifying, you compare each side term by term:- The coefficients on the left side must equal the sum of the coefficients on the right.
The result is a series of simpler, smaller equations forming a system that looks like: - For the \( x^3 \) terms - For the \( x^2 \) terms - For the \( x \) terms - For the constant termsSolving this system using techniques like substitution or elimination provides the particular values for \( A, B, C, \) and \( D \). This is crucial because each coefficient directly affects the outcome of the original rational expression when broken into finer parts. Understanding how to solve these systems is key to mastering partial fraction decompositions.