Problem 65 - Calculus Single Variable

Question

The correct partial fraction decomposition of \(\frac{5 x^{2}+18 x-1}{(x+4)^{2}(x-3)}\) is of the form \(\frac{A_{1}}{x+4}+\frac{A_{2}}{(x+4)^{2}}+\frac{B}{x-3}\) The values of the coefficients \(A_{1}, A_{2},\) and \(B\) are determined in Example \(6 .\) In Exercises 65 and \(66,\) show that the given incorrect form does not admit a solution for the unknown coefficients. $$ \frac{A_{1}}{x+4}+\frac{B}{x-3} $$

Step-by-Step Solution

Answer

The incorrect form \(\frac{A_{1}}{x+4}+\frac{B}{x-3}\) omits necessary terms and can't match the given numerator.

Step 1: Identify the Denominator Components

The denominator given in the problem is \((x+4)^2(x-3)\). Since one factor \((x+4)\) is repeated, the correct decomposition should include terms for both \(\frac{1}{x+4}\) and \(\frac{1}{(x+4)^2}\), in addition to \(\frac{1}{x-3}\). Thus, the form \(\frac{A_1}{x+4} + \frac{B}{x-3}\) misses \(\frac{A_2}{(x+4)^2}\).

Step 2: Write the Improper Form for Partial Fractions

The problem incorrectly tries to express the function as \(\frac{A_1}{x+4} + \frac{B}{x-3}\). To show this won't work, assume this form is valid, where we need to equate it to \(\frac{5x^2 + 18x - 1}{(x+4)^2(x-3)}\).

Step 3: Find a Common Denominator

For the improper form \(\frac{A_1}{x+4} + \frac{B}{x-3}\), the common denominator would be \((x+4)(x-3)\). Therefore, \(\frac{A_1}{x+4} + \frac{B}{x-3} = \frac{A_1(x-3) + B(x+4)}{(x+4)(x-3)}\).

Step 4: Equate Numerators

To equate the numerators, set \(A_1(x-3) + B(x+4) = 5x^2 + 18x - 1\). Expand the left side to \((A_1 + B)x + (4B - 3A_1)\).

Step 5: Compare Coefficients

Compare the coefficients of corresponding powers of \(x\): 1. \(x^2\) term: None on left vs 5 on right: This implies a contradiction, as no quadratic term exists in A. 2. \(x\) term: \((A_1 + B) = 18\) 3. Constant term: \(4B - 3A_1 = -1\) Verify that no solution for \(A_1\) and \(B\) satisfies these conditions due to the absence of \(x^2\) term.

Step 6: Conclusion: No Solution Exists

The important realization here is that because the quadratic term on the left is zero and on the right is 5, it's impossible to find values of \(A_1\) and \(B\) that satisfy the given equations. Therefore, the incorrect form \(\frac{A_1}{x+4} + \frac{B}{x-3}\) does not allow for a valid partial fraction decomposition.

Key Concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. These types of functions are central to understanding partial fraction decomposition, especially when working with polynomial equations in calculus and algebra.
For example, a rational function looks like \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\). In our exercise, we are dealing with \( \frac{5x^2 + 18x - 1}{(x+4)^2(x-3)} \), where the numerator \(5x^2 + 18x - 1\) and the denominator \((x+4)^2(x-3)\) are both polynomials.

To effectively decompose this rational function, it's essential to understand its structure and how different polynomial factors in the denominator dictate the form of the partial fraction. This understanding helps in splitting a complex rational term into simpler fractions, aiding in further calculations or integrations.

• Ensure the denominator and numerator are polynomials.
• The denominator must not equal zero.
• The factorization plays a crucial role in partial fraction decomposition.

Polynomial Equations

Polynomial equations form the backbone of rational functions and are pivotal in our study of partial fraction decomposition. Polynomials are expressions consisting of variables and coefficients organized in terms. They can often be solved to find roots or simplified by decomposition.
In the context of this exercise, you're looking at a denominator that requires understanding polynomial roots and powers. The denominator, \((x+4)^2(x-3)\), features both repeated and linear factors.
Repeated factors, such as \((x+4)^2\), call for special attention since they necessitate additional terms in the partial fraction decomposition. Recognizing and handling these correctly is key. Thus, careful factorization of the polynomial equation in the denominator is crucial both for proper setup and later solving.

• Identify repeated factors as these affect your decomposition terms.
• Ensure all linear factors are acknowledged for decomposition.
• Understand the root and degree of each polynomial term.

Comparing Coefficients

Comparing coefficients is a vital technique when you're solving for unknowns in polynomial decompositions. It involves setting up identity equations by aligning equivalent powers from both sides of the equation, derived from the common denominator.
In the case of our partial fraction decomposition for \( \frac{5x^2 + 18x - 1}{(x+4)^2(x-3)} \) being incorrectly written as \( \frac{A_1}{x+4} + \frac{B}{x-3} \), we need to compare coefficients to discover why this form fails.

Since the correct form is missing a term \( \frac{A_2}{(x+4)^2} \), comparing coefficients shows a mismatch. Here's the step-by-step:

• The anticipated quadratic term is missing as no \(x^2\) term can be created on the left side; thus, \(A_1(x-3) + B(x+4) = 5x^2 + 18x - 1\) can never equal this.


• The comparison reveals that symmetrical powers (like quadratic, linear, constant) must exist on both sides.
• Values are solved by equating like terms, but with missing terms, equations become unsolvable.

This structural comparison underlines why using the incorrect decomposition won't yield any rightful solutions for any real coefficients \(A_1\) and \(B\).