Problem 15 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{4}{2 x^{2}-5 x-3}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given expression is \(\frac{4}{2x^{2}-5x-3} = \frac{4}{2x+1} + \frac{2}{x-3}\)

Step 1: Factor the Denominator

First, factor the denominator of the given rational expression, \(2x^{2} - 5x - 3\). This can be factored to \(2x^2 - 6x + x - 3\), and eventually to \(2x(x-3) + 1(x-3)\) and finally to \((2x+1)(x-3)\).

Step 2: Set up the Partial Fractions

To set up the partial fractions, write the given rational expression as a sum of simpler fractions with unknown coefficients, \(A\) and \(B\), as follows: \(\frac{4}{2x^{2}-5x-3} = \frac{A}{2x+1} + \frac{B}{x-3}\).

Step 3: Find Coefficients

By multiplying both sides by the denominator from the left side to clear the fractions, the expression becomes \(4 = A(x-3) + B(2x+1)\). Collect like terms and then find the roots of the equation by setting each factor equal to zero. From the equation \(4=A(x-3)+2Bx+B\), it can be noticed that for \(x=3\), \(B=2\) and for \(x=-\frac{1}{2}\), \(A=4\).

Step 4: Write out the Decomposition

Lastly, substituting coefficients \(A\) and \(B\) back to the simpler fractions generates the complete partial fraction decomposition: \(\frac{4}{2x^{2}-5x-3} = \frac{4}{2x+1} + \frac{2}{x-3}\)

Key Concepts

Rational Expressions

Rational expressions are fractions that involve variables and polynomials in both their numerators and denominators. Just like numerical fractions, they can be simplified, added, subtracted, multiplied, and divided if proper algebraic rules are followed.

To understand rational expressions better, let's think of them as mathematical sentences that describe a relationship between two quantities. The denominator describes how many parts we divide a whole into, while the numerator describes how many parts we are considering. When variables are involved, this relationship can change depending on the values of those variables.

Simplifying rational expressions often involves factoring polynomials, as seen in the partial fraction decomposition exercise. It's a key skill that can help students solve complex algebra problems, integrate functions, and even solve differential equations.

Factoring Polynomials

Factoring polynomials is a process of breaking down a complex expression into simpler terms or factors that, when multiplied together, give back the original polynomial. It's a fundamental technique in algebra which allows us to simplify expressions, solve polynomial equations, and perform operations like partial fraction decomposition.

Typically, factoring starts with identifying the greatest common factor among the terms, then applying various techniques such as grouping, the difference of squares, sum/difference of cubes, or the use of special factoring formulas. Mastery of factoring offers a path to understand and solve a wide range of mathematical problems and is indispensable in calculus and beyond.

Example of Factoring:

• For the polynomial \(2x^{2} - 5x - 3\), it factors into \((2x + 1)(x - 3)\), providing two simpler expressions.

Recognizing patterns and applying the correct factoring technique is a skill developed through practice.

Algebraic Fractions

Algebraic fractions, also known as fractional expressions, are fractions where the numerator, denominator, or both contain algebraic expressions. These can pose significant challenges as one moves to more advanced algebra and calculus.

Working with algebraic fractions often involves simplifying, finding common denominators, and canceling like terms. Furthermore, partial fraction decomposition, used in integration, is a method of breaking down complex algebraic fractions into simpler parts that are easier to integrate or understand.

Handling Complex Fractions:

• Clearing complex denominators by multiplying through by the least common multiple (LCM).
• Becoming comfortable with operations on monomial, binomial, and more complex denominators.

Once mastered, algebraic fractions enable students to tackle a broader range of problems, making them a critical area in the journey of learning mathematics.