Problem 1 - College Algebra

Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of \( \frac{11 x - 10}{(x - 2) (x + 1)} \) is \( \frac{A}{x-2} + \frac{B}{x+1} \).

Step 1: Identify the denominator's factors

The denominator of the given rational expression is \( (x - 2)(x + 1) \). It means that its roots are 2 and -1.

Step 2: Write the general form of partial fraction

In a partial fraction decomposition, each denominator root corresponds to its own fraction. In this case, the roots are 2 and -1, which correspond to the denominators \( (x-2) \) and \( (x+1) \) respectively. Hence, the general form of a partial fraction is \( \frac{A}{x-2} + \frac{B}{x+1} \), where A and B are constants to be determined if we were to fully decompose this fraction.

Key Concepts

Understanding Rational Expression

A rational expression is quite similar to a fraction, except that both its numerator and denominator are polynomials. In simpler terms, it is the ratio of two polynomials, one on top and one on the bottom. For an expression like \(\frac{11x - 10}{(x-2)(x+1)}\), the numerator is \(11x - 10\) and the denominator is \((x-2)(x+1)\). Rational expressions are useful in breaking down complex algebraic problems. They allow us to simplify expressions and solve them more easily by factoring and simplifying each part of the expression separately.

• Numerator: The polynomial on top of the rational expression.
• Denominator: The polynomial on the bottom, which can't be zero.

It's crucial to keep the denominator factorized, as this helps in the decomposition process explained below.

Exploring the Denominator

The denominator of a rational expression plays a vital role in partial fraction decomposition. For \(\frac{11x - 10}{(x-2)(x+1)}\), the denominator \((x-2)(x+1)\) features two distinct linear factors.Identifying these factors is the first step when performing a partial fraction decomposition, a process necessary to express the rational expression as a sum of simpler fractions. - The denominator helps determine the form each part should take in the partial fraction.- It must be factorized completely to identify possible roots.If the denominator polynomials had repeated factors, that consideration would add extra complexity, but in this case, it's straightforward.

Finding the Roots of Polynomial

Root-finding is key in partial fraction decomposition. The roots are essentially the solutions to the equation when the polynomial equals zero. For our rational expression, the denominator \((x-2)(x+1)\) is essential in determining those roots.Solving \((x-2)(x+1)=0\) yields the roots:

• \(x = 2\)
• \(x = -1\)

These roots correspond to the values that make each factor zero. In this context, they dictate the form of the fractions in the decomposition. Each distinct root gives rise to a separate term in the partial fraction expansion.

Role of Constants in Fractions

In a partial fraction decomposition, the critical step is solving for the unknown constants in each part of the decomposed expression. These constants, noted as \(A\) and \(B\) in the decomposition \(\frac{A}{x-2} + \frac{B}{x+1}\), are what you'll find once the partial fraction is fully solved.While our exercise does not require solving for \(A\) or \(B\), understanding their significance is crucial. These constants adjust each fraction to accurately reproduce the original rational expression when combined.

• These constants help bridge the gap between the simplified fractions and the original expression.
• Finding them involves equating coefficients or substituting values to solve for \(A\) and \(B\).

By setting up the rational expression this way, we prepare the framework needed to delve deeper into algebraic solutions.