Problem 44
Question
Find the decomposition of the partial fraction for the irreducible repeating quadratic factor. \(\frac{3 x^{3}+2 x^{2}+14 x+15}{\left(x^{2}+4\right)^{2}}\)
Step-by-Step Solution
Verified Answer
\(\frac{3x + 2}{x^2 + 4} + \frac{2x + 7}{(x^2 + 4)^2}\)
1Step 1: Identify the form of partial fraction decomposition
The expression has a denominator with a repeated irreducible quadratic factor \((x^2+4)^2\). For such cases, the partial fraction decomposition will have the form: \(\frac{Ax+B}{x^2+4} + \frac{Cx+D}{(x^2+4)^2}\).
2Step 2: Set up the equation for decomposition
Express the rational function as the sum of its decomposition: \[\frac{3x^3 + 2x^2 + 14x + 15}{(x^2 + 4)^2} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2}\] Multiply through by \((x^2 + 4)^2\) to eliminate the denominators.
3Step 3: Simplify and equate terms
After clearing the denominators, expand and simplify both sides of the equation to obtain: \[3x^3 + 2x^2 + 14x + 15 = (Ax + B)(x^2 + 4) + (Cx + D)\] This will give: \[3x^3 + 2x^2 + 14x + 15 = Ax^3 + 4Ax + Bx^2 + 4B + Cx + D\]
4Step 4: Group like terms and solve for coefficients
Arrange like terms on both sides of the equation:- The coefficient of \(x^3\) gives \(A = 3\).- The coefficient of \(x^2\) gives \(B = 2\).- The coefficient of \(x\) gives \(4A + C = 14\), solving for \(C\), \(C = 14 - 4(3) = 2\).- The constant term gives \(4B + D = 15\), solving for \(D\), \(D = 15 - 4(2) = 7\).
5Step 5: Write the partial fraction decomposition
Substitute the values \(A = 3\), \(B = 2\), \(C = 2\), and \(D = 7\) into the decomposition form to obtain: \[\frac{3x + 2}{x^2 + 4} + \frac{2x + 7}{(x^2 + 4)^2}\].
Key Concepts
Irreducible Quadratic FactorRepeated FactorsRational FunctionAlgebraic Expressions
Irreducible Quadratic Factor
In algebra, an irreducible quadratic factor is a quadratic expression that cannot be factored further using real numbers. For example, the expression \(x^2 + 4\) is irreducible because it doesn't have real roots that can simplify it into linear factors.
When dealing with a rational function that involves irreducible quadratic factors, like in the given original exercise, these quadratics can complicate partial fraction decomposition. To handle them, our decomposition includes terms of the form \(\frac{Ax + B}{x^2 + 4}\), indicating that the numerator should be a linear expression.
Understanding irreducible quadratic factors is crucial because it guides how to set up the decomposition for rational functions, especially when they appear as repeated factors in the denominator.
When dealing with a rational function that involves irreducible quadratic factors, like in the given original exercise, these quadratics can complicate partial fraction decomposition. To handle them, our decomposition includes terms of the form \(\frac{Ax + B}{x^2 + 4}\), indicating that the numerator should be a linear expression.
Understanding irreducible quadratic factors is crucial because it guides how to set up the decomposition for rational functions, especially when they appear as repeated factors in the denominator.
Repeated Factors
Repeated factors in the denominator of a rational function indicate that a particular root or expression occurs more than once. In partial fraction decomposition, this requires special attention since each occurrence influences the form of the decomposition.
For example, take the factor \((x^2 + 4)^2\). This is a repeated irreducible quadratic factor. It appears as the square of \(x^2 + 4\). In our decomposition, we need multiple terms to account for each repetition:
Handling repeated factors correctly in algebraic expressions is crucial for achieving an accurate decomposition, so they are an essential aspect of learning and applying the partial fraction decomposition technique.
For example, take the factor \((x^2 + 4)^2\). This is a repeated irreducible quadratic factor. It appears as the square of \(x^2 + 4\). In our decomposition, we need multiple terms to account for each repetition:
- \(\frac{Ax + B}{x^2 + 4}\)
- \(\frac{Cx + D}{(x^2 + 4)^2}\)
Handling repeated factors correctly in algebraic expressions is crucial for achieving an accurate decomposition, so they are an essential aspect of learning and applying the partial fraction decomposition technique.
Rational Function
A rational function is essentially the ratio of two polynomials. The given example \(\frac{3x^3 + 2x^2 + 14x + 15}{(x^2 + 4)^2}\) is a prime instance of such a function.
Rational functions often pose challenges because the polynomial in the denominator (like our \((x^2 + 4)^2\)) can complicate simplification. However, understanding their structure is fundamental to many areas of algebra and calculus. When decomposing into partial fractions, the goal is to express the rational function as a sum of simpler fractions, aiding both integration and other algebraic manipulations.
Recognizing and correctly setting up the form of these simpler expressions helps manage and solve more complex rational expressions found in many mathematical applications.
Rational functions often pose challenges because the polynomial in the denominator (like our \((x^2 + 4)^2\)) can complicate simplification. However, understanding their structure is fundamental to many areas of algebra and calculus. When decomposing into partial fractions, the goal is to express the rational function as a sum of simpler fractions, aiding both integration and other algebraic manipulations.
Recognizing and correctly setting up the form of these simpler expressions helps manage and solve more complex rational expressions found in many mathematical applications.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). In the context of our exercise, both the numerator \(3x^3 + 2x^2 + 14x + 15\) and the denominator \((x^2 + 4)^2\) form such expressions.
Mastering algebraic expressions is vital for manipulating rational functions and achieving correct partial fraction decompositions. It involves understanding how to expand expressions, combine like terms, and solve for coefficients.
In the step-by-step solution, we expanded \((Ax + B)(x^2 + 4) + (Cx + D)\) and arranged terms according to powers of \(x\), which requires comfort with algebraic manipulation. This process highlights the importance of being adept at simplifying and handling complex algebraic expressions to solve deeper mathematical problems efficiently.
Mastering algebraic expressions is vital for manipulating rational functions and achieving correct partial fraction decompositions. It involves understanding how to expand expressions, combine like terms, and solve for coefficients.
In the step-by-step solution, we expanded \((Ax + B)(x^2 + 4) + (Cx + D)\) and arranged terms according to powers of \(x\), which requires comfort with algebraic manipulation. This process highlights the importance of being adept at simplifying and handling complex algebraic expressions to solve deeper mathematical problems efficiently.
Other exercises in this chapter
Problem 44
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