Problem 4 - Precalculus Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x}{x^{2}+3 x-4}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{A}{x+4} + \frac{B}{x-1} \).

Step 1: Identify type of denominator

The denominator of the rational function is a quadratic polynomial, specifically a second-degree polynomial, given by \(x^2 + 3x - 4\).

Step 2: Factor the denominator

Factor the quadratic expression \(x^2 + 3x - 4\) into linear factors. This can be factored as \((x + 4)(x - 1)\).

Step 3: Set up the partial fraction decomposition form

Since the denominator factors into distinct linear factors \((x+4)(x-1)\), the partial fraction decomposition will be set up with two separate fractions. Each fraction will have a denominator corresponding to one of the linear factors. Let \(A\) and \(B\) be constants to find later:\[\frac{x}{x^2 + 3x - 4} = \frac{A}{x+4} + \frac{B}{x-1}\]

Step 4: Combine the fractions

Combine the expressions on the right side over a common denominator:\[\frac{A}{x+4} + \frac{B}{x-1} = \frac{A(x-1) + B(x+4)}{(x+4)(x-1)}\]

Step 5: Equate numerators

Since the original fraction \(\frac{x}{x^2 + 3x - 4}\) must be equivalent to \(\frac{A(x-1) + B(x+4)}{(x+4)(x-1)}\), equate the numerators:\[x = A(x-1) + B(x+4)\]

Key Concepts

Rational Functions

Rational functions are expressions that involve the division of two polynomials. In simpler terms, a rational function is a ratio of two polynomial functions, where the numerator and the denominator are both polynomials.
They are mathematically represented as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). Understanding rational functions is key to mastering partial fraction decomposition because this process is all about breaking down complex rational expressions into simpler parts.

• The degree of the polynomial in the numerator must be less than the degree of the polynomial in the denominator for the expression to be a proper rational function.
• If the degree of the numerator is greater, we first perform polynomial long division to simplify the expression.

In our example, \( \frac{x}{x^2+3x-4} \), we notice that the numerator \( x \) is a polynomial of degree 1, and the denominator \( x^2+3x-4 \) is of degree 2, forming a proper rational function.

Quadratic Polynomials

Quadratic polynomials are polynomial expressions of degree 2, generally written in the form \( ax^2 + bx + c \). These types of polynomials can have various characteristics that need to be understood for decomposition.

• They can often be factored into two linear factors, particularly if they have real and distinct roots.
• When factoring, the quadratic formula or factoring by inspection may be used to find these roots.

In the given example, the quadratic polynomial \( x^2 + 3x - 4 \) appears in the denominator.
By factoring this quadratic polynomial, we find the linear factors \( (x+4) \) and \( (x-1) \). Recognizing that a quadratic polynomial can often be reduced to linear factors is a crucial step in partial fraction decomposition, making it easier to rewrite complex expressions.

Linear Factors

Linear factors are simpler polynomial expressions of degree 1, such as \( x + a \). In partial fraction decomposition, factoring a polynomial into linear factors allows us to express the original rational function as a sum of simpler fractions.
These fractions are easier to work with because they involve unknown constants that can be solved for, using algebraic techniques.

• Each distinct linear factor in the denominator of the original rational function leads to a separate fraction in the decomposition.
• The partial fraction decomposition will have the form \( \frac{A}{x+a} \) for each factor \( x+a \).

In our example, after factoring the quadratic polynomial \( x^2+3x-4 \), we use the linear factors \( (x+4) \) and \( (x-1) \) to set up the partial fraction decomposition: \[ \frac{A}{x+4} + \frac{B}{x-1} \].
This transformation makes solving complex integrals or solving differential equations more straightforward, as algebraic manipulation and integration become simpler with linear terms.