Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Determine whether the partial fraction decomposition of the first expression is the second expression. See Example 1. $$\frac{3 x-1}{x^{2}-x} ; \frac{1}{x}+\frac{3}{x-1}$$

Step-by-Step Solution

Answer

No, the partial fraction decomposition is not the same.

Step 1: Factor the Denominator

The denominator of the fraction \( \frac{3x-1}{x^2-x} \) is \( x^2 - x \). Factor it to get \( x(x-1) \). This will help us in writing the partial fraction decomposition.

Step 2: Set Up Partial Fractions

Write the expression \( \frac{3x-1}{x(x-1)} \) as a sum of partial fractions with unknown coefficients: \( \frac{A}{x} + \frac{B}{x-1} \).

Step 3: Combine Partial Fractions

Combine the fractions on the right-hand side: \( \frac{A(x-1) + Bx}{x(x-1)} \). Since the denominators are the same, we can equate the numerators.

Step 4: Equate and Solve for Coefficients

Equate \( 3x - 1 \) to \( A(x-1) + Bx \). Expand and simplify: \( Ax - A + Bx = 3x - 1 \). Combine like terms to get \( (A+B)x - A = 3x - 1 \). This gives the system of equations: 1) \( A + B = 3 \), 2) \( -A = -1 \).

Step 5: Solve for A and B

From equation 2, \( -A = -1 \) gives \( A = 1 \). Substitute into equation 1 to get \( 1 + B = 3 \), which gives \( B = 2 \).

Step 6: Substitute Coefficients Back

Substitute \( A = 1 \) and \( B = 2 \) back into the partial fraction expression: \( \frac{1}{x} + \frac{2}{x-1} \).

Step 7: Compare with Given Expression

Compare \( \frac{1}{x} + \frac{2}{x-1} \) with the given expression \( \frac{1}{x} + \frac{3}{x-1} \). The two expressions are not the same because the coefficients of \( \frac{1}{x-1} \) differ.

Key Concepts

Fraction Decomposition

Fraction decomposition, also known as partial fraction decomposition, is a method used to express complex rational expressions as a sum of simpler fractions. This technique is particularly useful when dealing with integrals involving rational expressions or when simplifying algebraic expressions.

To perform fraction decomposition:

• First, ensure the degree of the numerator is less than the degree of the denominator. If not, perform long division to simplify.
• Factor the denominator completely. This step is crucial as it sets the basis for breaking the fraction into parts.
• Write the original fraction as a sum of fractions, where each denominator is a factor from the factored form. Assign a variable, often denoted as A, B, C, etc., to each fraction's numerator.
• Combine these fractions back into one, matching the original denominator.

After setting up the equation, compare coefficients to find the values of variables, turning the decomposed form into an expression equivalent to the original one.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions behave similarly to numerical fractions, but operations involve variable expressions.

Key aspects about rational expressions include:

• Simplification: Simplify by factoring both numerator and denominator and canceling any common factors. This is similar to simplifying numerical fractions by reducing them.
• Operations: Addition, subtraction, multiplication, and division of rational expressions follow the same rules as fractions. Ensure denominators are the same when adding or subtracting; this may involve finding a common denominator.
• Domain: Determine the values of variables for which the expression is undefined. This usually occurs where the denominator equals zero.

Rational expressions are a fundamental topic in algebra, laying the groundwork for solving equations involving these more complex forms.

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions, or factors, that when multiplied give the original expression. It's a pivotal skill in algebra, aiding in simplifying expressions and solving equations.

Here are some basic types of factoring:

• Common Factor: Identify a common factor in all terms and factor it out.
• Trinomial: Use the trial-and-error method or formulas to factor trinomials of the form \(ax^2 + bx + c\).
• Difference of Squares: Recognize and factor expressions like \(a^2 - b^2\) into \((a + b)(a - b)\).
• Grouping: For expressions with four or more terms, use grouping to simplify.

Factoring turns what might be complex algebraic equations into simpler multiplication problems, making it easier to solve for variables or rewrite expressions in a more workable format.