Problem 63 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Step-by-Step Solution

Answer

No, the statement does not make sense because for the given fraction with repeated linear factor \((x+3)^2\), correct decomposition should have been \( \frac{A}{x+3} + \frac{B}{(x+3)^2} \) instead of \( \frac{A}{x+3} + \frac{B}{x+3} \).

Step 1: Understand the partial fraction decomposition

The partial fraction decomposition is a way to express the fraction as a sum of simpler fractions. If we have a repeated linear factor like \((x+3)^2\) in the denominator of a fraction, it is decomposed into fractions with \((x+3)\) and \((x+3)^2\) in the denominator.

Step 2: Analyze the given decomposition

The given partial fraction decomposition is: \( \frac{5x+2}{(x+3)^2} = \frac{A}{x+3} + \frac{B}{x+3} \). Here we can notice that instead of \(B) /(x+3)^2\), \(B /(x+3)\) is wrongly written in the fraction.

Step 3: Conclude the correctness of the statement

Based on the standard method, we can see that the given statement didn't decompose the fraction correctly, therefore it doesn't make sense.

Key Concepts

Repeated Linear Factor

When dealing with the partial fraction decomposition of rational expressions, repeated linear factors in the denominator must be handled carefully. A linear factor is an expression like \(x + 3\). If this factor is repeated, appearing as \(x + 3)^2\), it means it occurs twice in the factorization. To properly decompose a rational expression with a repeated factor, you must include terms for each power of that factor in the denominator.

For example, with the denominator \(x + 3)^2\), the decomposition must include:

• \( \frac{A}{x+3} \)
• \( \frac{B}{(x+3)^2} \)

This approach ensures that each occurrence of the factor is accounted for, making it possible to combine fractions to return to the original expression.

Algebraic Fractions

Algebraic fractions are expressions that contain polynomials in the numerator and the denominator. They act similarly to numerical fractions but include variables. Decomposing algebraic fractions into partial fractions simplifies complex expressions into a sum of simpler ones.

To work with algebraic fractions:

• Ensure the polynomial in the numerator has a lower degree than in the denominator. If not, perform polynomial long division first.
• Recognize factors in the denominator to set up the correct partial fraction decomposition.
• Use separate terms for different types of factors, such as linear and quadratic.

This simplification is particularly useful in calculus and differential equations, making it easier to integrate or differentiate complex expressions.

Rational Expressions

Rational expressions are quotients of two polynomials, similar to fractions but involving variables. Understanding these expressions is crucial for algebra, calculus, and beyond. Rational expressions require manipulation to simplify them, and one common method is partial fraction decomposition.

Key points about rational expressions include:

• They must be simplified by factoring and reducing similar to numerical fractions.
• Partial fraction decomposition helps express them as sums of simpler fractions, advantageous for integration.
• Correct decomposition accounts for all factors, especially repeated ones, to rebuild the original fraction accurately.

Mastering rational expressions and their decomposition deepens your understanding of algebraic relationships and prepares you for more advanced mathematical topics.