Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{x}{(x-1)^{2}}+\frac{2}{x}-\frac{x+1}{x^{3}-x^{2}} $$

Short answer

\(\frac{3x^3 - 5x^2 + 2x + 1}{x^2 (x-1)^2}\)

Step-by-step solution

- Find the common denominator

The denominators are \((x-1)^2\), \(x\), and \(x^3 - x^2 = x^2(x-1)\). The common denominator is \(x^2(x-1)^2\).

- Rewrite each fraction with the common denominator

\[ \frac{x}{(x-1)^2} = \frac{x \times x^2}{(x-1)^2 \times x^2} = \frac{x^3}{x^2 (x-1)^2} \ \frac{2}{x} = \frac{2 \times x(x-1)^2}{x \times x^2(x-1)^2} = \frac{2x(x-1)^2}{x^2 (x-1)^2} \ \frac{x+1}{x^3 - x^2} = \frac{x+1}{x^2 (x-1)} = \frac{(x+1) \times (x-1)}{x^2 (x-1) \times (x-1)} = \frac{(x+1)(x-1)}{x^2 (x-1)^2} = \frac{x^2-1}{x^2 (x-1)^2}\]

- Combine the fractions

\[ \frac{x^3}{x^2 (x-1)^2} + \frac{2x(x-1)^2}{x^2 (x-1)^2} - \frac{x^2-1}{x^2 (x-1)^2}\]

- Simplify the numerator

\[ \frac{x^3 + 2x(x-1)^2 - (x^2-1)}{x^2 (x-1)^2} = \frac{x^3 + 2x (x^2 - 2x + 1) - x^2 + 1}{x^2 (x-1)^2} = \frac{x^3 + 2x^3 - 4x^2 + 2x - x^2 + 1}{x^2 (x-1)^2} = \frac{3x^3 - 5x^2 + 2x + 1}{x^2 (x-1)^2}\]

- Factor the numerator, if possible

Attempt to factor the numerator \(3x^3 - 5x^2 + 2x + 1\). It does not factor nicely, so leave it as is.

- Express the final answer in factored form if possible

\[ \frac{3x^3 - 5x^2 + 2x + 1}{x^2 (x-1)^2}\]

Key concepts

Common Denominator

To add or subtract rational expressions, we need a common denominator. A common denominator is a shared multiple of the denominators of all the fractions involved. In this exercise, the denominators were \( (x-1)^2 \), \( x \), and \( x^3 - x^2 = x^2(x-1) \). We chose \( x^2(x-1)^2 \) because it contains all unique factors from each denominator, ensuring each fraction can be rewritten with this shared denominator.

Steps to find a common denominator:

• Factor each denominator completely.
• Identify all unique factors.
• Include the highest power of each factor from any of the denominators.

This ensures that the new denominators for each fraction align, allowing them to be combined easily.

Acquiring a common denominator helps us rewrite each fraction and combine them efficiently, which is crucial in simplifying the expression.

Factored Form

Factored forms are essential in algebra, especially in operations involving rational expressions. In the given exercise, the final expression is left in its factored form.

The factored form of an algebraic expression means expressing it as a product of its simplest units. For polynomials, these units are called factors. Here’s why factored form is necessary:

• It simplifies expressions.
• It makes multiplication and division easier.
• It helps identify common factors which can be canceled out.

In this exercise, we express our denominators in factored form first. For example, \( x^3 - x^2 \) becomes \( x^2 (x-1) \). This step is crucial for finding the common denominator.

Finally, after rewriting our fractions with the common denominator and combining them, we check to see if the numerator can be factored. Here, \( 3x^3 - 5x^2 + 2x + 1 \) doesn't factor nicely, so it is left as is.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions where the numerator and/or the denominator are polynomials. Working with these fractions involves operations similar to those with numerical fractions, including finding common denominators, addition, subtraction, multiplication, and division.

Key concepts for algebraic fractions:

• Simplification: Factor both numerator and denominator and cancel common factors.
• Operations: When adding or subtracting, find a common denominator. For multiplication and division, multiply/divide straight across.
• Restrictions: Identify values that make the denominator zero, as these are not allowed in algebraic fractions.

In this exercise, we dealt with three algebraic fractions with different denominators. By finding the common denominator and rewriting each fraction accordingly, they could be combined and simplified.

Mastering algebraic fractions equips students to handle more complex algebraic operations, facilitating their progression in mathematics.