Problem 3 - Precalculus

Question

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{m}{(7 x-6)\left(x^{2}+9\right)} $$

Step-by-Step Solution

Answer

\( \frac{A}{7x - 6} + \frac{Bx + C}{x^2 + 9} \) is the form for partial fraction decomposition.

Step 1: Identify the Denominator

The first step is to examine the denominator of the fraction. In this particular case, the denominator is \((7x - 6)(x^2 + 9)\). Here, we have a linear factor \(7x - 6\) and an irreducible quadratic factor \(x^2 + 9\).

Step 2: Determine Partial Fraction Form

Given that the denominator has a linear factor and an irreducible quadratic factor, the partial fraction decomposition will include one term for the linear factor and one term for the quadratic factor. The form is: \[ \frac{A}{7x - 6} + \frac{Bx + C}{x^2 + 9} \] where \(A\), \(B\), and \(C\) are constants to be determined.

Key Concepts

Rational Expressions

A rational expression is similar to a fraction, but instead of just numbers in the numerator and denominator, it includes polynomials. Think of a rational expression as a fraction where both the top and bottom are algebraic expressions rather than simple numbers.
These expressions can describe a wide range of mathematical scenarios, from basic arithmetic to more complex calculus problems involving variables.
When dealing with rational expressions, it's crucial to ensure that the denominator is not zero since division by zero is undefined. This is a common pitfall many learners encounter.
Steps involved in working with rational expressions typically include:

• Factorization: Breaking down polynomials into simpler factors. This makes it easier to identify common factors and simplify the expression.
• Simplification: Canceling out terms that appear both in the numerator and the denominator.
• Partial Fraction Decomposition: Writing the expression in a form where it's easier to analyze or integrate, especially useful in calculus.

Understanding rational expressions sets a strong foundation for working with equations that model real-world situations.

Linear Factor

A linear factor is a basic building block in algebra, appearing as polynomials of the first degree, meaning they are in the form of \( ax + b \).
This form is called a linear polynomial because it graphs as a straight line when plotted. In our exercise, the linear factor is \( 7x - 6 \).
Linear factors are crucial in partial fraction decomposition because they simplify one part of our expression. Here's why they are important:

• Easy to Work With: Linear factors can be quickly integrated or derivate, making them friendly for calculus operations.
• Main Term in Decomposition: In partial fraction decomposition, each linear factor in the denominator corresponds to a simpler term, aiding in breaking down complex expressions.
• Roots Insight: Solving the linear factor for zero gives its root, which is useful for graphing and solving for intersections.

By understanding linear factors, you unlock one of the keys to handling algebraic expressions, whether in basic math or advanced calculus.

Quadratic Factor

A quadratic factor is a polynomial of degree two, generally appearing as \( ax^2 + bx + c \).
Unlike a linear factor, a quadratic one represents a parabolic curve when graphed. In the provided exercise, \( x^2 + 9 \) is a quadratic factor.
Quadratic factors are often encountered in partial fraction decomposition where they play a different role from linear factors. Here are some insights:

• Can't Always Solve Directly: Often, quadratic factors are irreducible over the reals, meaning they can't be factored into linear components without involving complex numbers.
• Contribute to Complexity: When performing partial fractions, quadratic factors add depth, requiring terms like \( Bx + C \) in the numerator to account for all degrees.
• Graphical Insights: Quadratics' roots and vertex provide meaningful information, such as the motion of projectiles or the optimization of areas.

Recognizing and working with quadratic factors is crucial for understanding some more advanced algebraic concepts like conic sections and equations of motion.