Problem 15 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The decomposition of the fraction \(\frac{4}{2x^2-5x-3}\) into partial fractions is \(\frac{2}{2x + 1} - \frac{1}{x - 3}\).

Step 1: Factor the denominator

The first step is to factorize the denominator \(2x^2-5x-3\). This can be done by breaking the middle term of the given quadratic equation \(2x^2-5x-3\). Let's rewrite it as \(2x^2-6x+x-3\). This equals \((2x^2-6x) + (x-3)\) which simplifies to \(2x(x-3)+(x-3)\). This further simplifies to \((2x+1)(x-3)\). We now have \(\frac{4}{2x^2-5x-3} = \frac{4}{(2x+1)(x-3)}\).

Step 2: Express the fraction as the sum of partial fractions

We break down the rational expression into simpler fractions. The resulting expression will be \(\frac{4}{(2x+1)(x-3)} = \frac{A}{(2x + 1)} + \frac{B}{(x - 3)}\).

Step 3: Find the values of A and B

This can be done by expressing the sum of fractions on the right as a single fraction, equating it to the fraction on the left and comparing coefficients of like terms. Doing this we get, \(4 = A(x-3) + B(2x+1)\). Plugging in selected values for x to eliminate one of the variables. Substituting \(x =3\) in the equation give \(A = 2\) and substituting \(x = -0.5\) gives \(B = -1\).

Step 4: Write the final answer

Substitute A and B back into the expression from Step 2. The decomposition of the fraction \(\frac{4}{2x^2-5x-3}\) into partial fractions is then \(\frac{2}{2x + 1} - \frac{1}{x - 3}\).

Key Concepts

Rational Expression

Rational expressions are fractions in which the numerator and the denominator are both polynomials. The example in our exercise, \(\frac{4}{2x^2-5x-3}\), is such a rational expression. Understanding rational expressions is crucial because they appear frequently in algebra, calculus, and other areas of mathematics and science.

When working with rational expressions, it's important to be able to simplify them and understand their behavior. Simplification can involve factoring the polynomials, reducing the expression if possible, or decomposing them into simpler parts, which is what partial fraction decomposition is all about. Partial fraction decomposition is particularly useful when integrating rational expressions or solving equations containing them.

Factoring Quadratics

Factoring quadratics is a fundamental skill in algebra that involves finding two binomials that multiply together to give the original quadratic expression. In the exercise, we have the quadratic equation \(2x^2-5x-3\). The process of factoring it requires rewriting the equation in a way that it can be expressed as a product of two binomials.

Approach to Factoring

• Look for a common factor in the terms.
• Rewrite the quadratic in such a way that breaks up the middle term.
• Group terms to factor by grouping.
• Simplify to find the binomial factors.

This is exactly what was done in our step-by-step solution, ultimately resulting in the factors \((2x+1)(x-3)\). Factoring is a critical step because it lays the foundation for partial fraction decomposition.

Solving for Coefficients

Solving for coefficients in partial fraction decomposition involves finding the values for the variables (usually denoted by letters such as A, B, C, etc.) in the decomposed fractions. These coefficients determine the contribution of each partial fraction to the whole expression.

To find these coefficients, as shown in the provided exercise solution, we equate the decomposed fractions to the original fraction and compare coefficients on both sides of the equation. Sometimes we can find these values by choosing smart values for the variable \(x\), to simplify the equation and isolate the coefficients. Getting these correct is vital because they directly influence the accuracy of the resultant decomposed expression.

Algebraic Fractions

Algebraic fractions are fractions that contain variables in their numerators, denominators, or both. These are just like numerical fractions but instead of numbers, they involve algebraic expressions. The concept of algebraic fractions extends the idea of numerical fractions into algebra, and they behave under similar principles such as common denominators for addition and subtraction, and inversion for division.

Working with algebraic fractions often encompasses simplifying the fractions, performing arithmetic with them, and sometimes resolving complex expressions into more manageable parts, which is where partial fraction decomposition comes into play. Mastering algebraic fractions is essential, as they are pivotal components of higher-level mathematics, including calculus.