Problem 3
Question
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$
Step-by-Step Solution
Verified Answer
\( \frac{2x-3}{x(x^{2}+10)} = \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \)
1Step 1: Factorize the Denominator Polynomial
To begin with, the denominator polynomial of the rational expression needs to be factored. For the expression \( \frac{2x-3}{x^{3}+10x} \), the denominator \( x^{3}+10x \) can be factorized as \( x(x^{2}+10) \).
2Step 2: Formulate the Partial Fraction Decomposition
Following the factorisation, the partial fraction decomposition of the rational expression can be presented with unknown constants. Since the denominator \( x(x^{2}+10) \) has a linear term \( x \), and a quadratic term \( x^{2}+10 \), the partial fraction decomposition will include two fractions, each corresponding to these factors, i.e: \( \frac{2x-3}{x(x^{2}+10)} = \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \). As we are not required to find the values of the constants \( A \), \( B \), and \( C \), this is the final answer.
Key Concepts
Rational ExpressionsFactorizationPolynomial Denominators
Rational Expressions
A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding rational expressions is key to grasping many algebraic concepts. In the example given, \( \frac{2x-3}{x^{3}+10x} \), the expression is rational because the numerator \( 2x-3 \) and the denominator \( x^{3}+10x \) are polynomials.
One fundamental aspect of dealing with rational expressions is ensuring the denominator is not zero. This maintains the expression's validity. In our example, \( x \) cannot be zero, or it would make the denominator zero.
Rational expressions often require simplification or decomposition, especially when dealing with partial fraction decomposition. This involves breaking the expression into simpler parts to make calculations easier and more understandable. This is particularly useful in calculus and solving differential equations.
One fundamental aspect of dealing with rational expressions is ensuring the denominator is not zero. This maintains the expression's validity. In our example, \( x \) cannot be zero, or it would make the denominator zero.
Rational expressions often require simplification or decomposition, especially when dealing with partial fraction decomposition. This involves breaking the expression into simpler parts to make calculations easier and more understandable. This is particularly useful in calculus and solving differential equations.
Factorization
Factorization is a process used to break down expressions into simpler parts, usually by finding numbers or expressions that multiply together to produce the original expression. It is especially useful when dealing with polynomial expressions.
In the exercise example, the denominator \( x^{3}+10x \) is factored by identifying common factors and applying algebraic identities. Initially, note that \( x \) can be factored out, so we rewrite it as \( x(x^{2}+10) \).
This step is critical, as factoring helps simplify the expression, which allows easier manipulation and decomposition into partial fractions. When tackling complex algebra problems, factorization is often a vital first step.
In the exercise example, the denominator \( x^{3}+10x \) is factored by identifying common factors and applying algebraic identities. Initially, note that \( x \) can be factored out, so we rewrite it as \( x(x^{2}+10) \).
This step is critical, as factoring helps simplify the expression, which allows easier manipulation and decomposition into partial fractions. When tackling complex algebra problems, factorization is often a vital first step.
Polynomial Denominators
A polynomial denominator is the bottom part of a fraction that consists of a polynomial. It can significantly influence how the entire expression behaves and is evaluated.
Partial fraction decomposition specifically needs examining the polynomial denominator to determine how the expression can be broken down into simpler partial fractions. For the given expression \( \frac{2x-3}{x^{3}+10x} \), its polynomial denominator \( x^{3}+10x \), once factored, is \( x(x^{2}+10) \).
The type of terms in this polynomial, such as linear \( x \) and quadratic \( x^2+10 \), guide how the rational expression should be decomposed into partial fractions. Here, having a mix of distinct linear and quadratic terms suggests a structure with constants \( A \), \( B \), and \( C \) as in: \( \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \). This breakdown simplifies complex calculations and integrals associated with rational expressions.
Partial fraction decomposition specifically needs examining the polynomial denominator to determine how the expression can be broken down into simpler partial fractions. For the given expression \( \frac{2x-3}{x^{3}+10x} \), its polynomial denominator \( x^{3}+10x \), once factored, is \( x(x^{2}+10) \).
The type of terms in this polynomial, such as linear \( x \) and quadratic \( x^2+10 \), guide how the rational expression should be decomposed into partial fractions. Here, having a mix of distinct linear and quadratic terms suggests a structure with constants \( A \), \( B \), and \( C \) as in: \( \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \). This breakdown simplifies complex calculations and integrals associated with rational expressions.
Other exercises in this chapter
Problem 3
Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x $$
View solution Problem 3
In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 3} \frac{2(x-3)}{x^{2
View solution Problem 4
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} \ln \left(x^{2}\right) d x $$
View solution Problem 4
In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{2
View solution