Problem 16 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

\(\frac{x}{x^{2}+2 x-3} = \frac{1/4}{x-1} + \frac{1/4}{x+3}\)

Step 1: Factor the denominator

Start by factoring the quadratic in the denominator \(x^{2}+2x-3\). One can use the standard method for factoring quadratics, which is to find two numbers that multiply to equal -3 (the constant term) and add to equal 2 (the coefficient of \(x\)). The factors are \(x-1\) and \(x+3\). So, the expansion is \(x^{2}+2x-3 = (x-1)(x+3)\).

Step 2: Split the fraction

The next step involves separating the given fraction into two smaller fractions, representing the initial expression as sum or differences of separate fractions. Each fraction will have each of the factors found in step 1 as denominator. It is known as the partial fraction decomposition. This gives \( \frac{x}{(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3} \), where \(A\) and \(B\) are constants to be determined.

Step 3: Solve for constants A and B

To find the constants \(A\) and \(B\), multiply both sides by the common denominator \((x-1)(x+3)\) to cancel out the denominator on the left side. This results in: \( x = A(x+3) + B(x-1) \). Now choose suitable values for \(x\) to make expression easier to solve. Choosing for instance \(x=1\) and \(x=-3\), one can solve for \(A\) and \(B\). Doing that, it is obtained that \(A = 1/4\) and \(B = 1/4\).

Step 4: Write final partial fraction decomposition

The constants \(A\) and \(B\) can now be substituted back into the partial fraction decomposition from step 2. This gives the final answer to the question: \( \frac{x}{x^{2}+2 x-3} = \frac{1/4}{x-1} + \frac{1/4}{x+3} \).

Key Concepts

Factoring quadratics

Understanding how to factor quadratics is a pivotal skill in algebra, essential for simplifying many types of expressions and solving quadratic equations. A quadratic is a polynomial of degree two, typically written in the form \( ax^2 + bx + c \). Factoring means to write the quadratic as a product of two binomials.

To factor a quadratic, one must find two numbers that multiply to the constant term, \(c\), and add to the coefficient of the \(x\) term, \(b\). In the exercise \( x^{2}+2x-3 \), the numbers that fulfil these requirements are 1 and -3. Thus, the factored form of \( x^{2}+2x-3 \) is \( (x-1)(x+3) \).

Once a quadratic is factored, it can be used to simplify rational expressions, solve for zeros of functions, and is especially useful in the process of partial fraction decomposition which breaks down complex fractions into simpler, more manageable pieces.

Rational expressions

What Are Rational Expressions?

Just like rational numbers are ratios (or fractions) of integers, rational expressions are ratios of polynomials. They have the form \( \frac{P(x)}{Q(x)} \) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \eq 0\).

Simplifying Rational Expressions

Similar to simplifying fractions by canceling common factors in the numerator and denominator, simplifying rational expressions involves factoring polynomials and reducing them to their simplest form. This can help in solving problems, graphing functions, and calculus operations like integration.

In our exercise, the rational expression \( \frac{x}{x^{2}+2x-3} \) is simplified through partial fraction decomposition after factoring the quadratic in the denominator.

Algebraic fractions

Understanding Algebraic Fractions

Algebraic fractions are fractions in which the numerator and/or the denominator are algebraic expressions. Just as with numerical fractions, the goal with algebraic fractions is often to simplify them or perform operations such as addition, subtraction, multiplication, and division.

When it comes to decomposing algebraic fractions into partial fractions, like in our example \( \frac{x}{(x-1)(x+3)} \), the process helps us to split a complex fraction into simpler parts – potentially easing the way to integrate or solve equations.

The Role of Partial Fractions

Partial fraction decomposition is particularly useful when dealing with integrals in calculus or making complex fraction multiplication and division more accessible by breaking it down into a sum of simpler fractions, just as demonstrated in the given solution by turning the complex fraction into the sum \( \frac{1/4}{x-1} + \frac{1/4}{x+3} \).