Problem 28 - Algebra 2 and Trigonometry

Question

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \left(x^{2}-2 x+1\right) \div \frac{x-1}{3} \cdot \frac{x+4}{3 x} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x^2 + 3x - 4}{x} \), undefined for \( x = 0 \) and \( x = 1 \).

Step 1: Simplify the Division

Start by simplifying the division operation. The expression \( (x^2 - 2x + 1) \div \frac{x-1}{3} \) can be rewritten using multiplication by the reciprocal: \( (x^2 - 2x + 1) \times \frac{3}{x-1} \). The expression \( x^2 - 2x + 1 \) is a perfect square trinomial and can be factored to \( (x-1)^2 \). So, the expression becomes \( (x-1)^2 \times \frac{3}{x-1} \).

Step 2: Simplify the Multiplication

Now simplify the multiplication: \( (x-1)^2 \times \frac{3}{x-1} \cdot \frac{x+4}{3x} \). First, simplify \( (x-1)^2 \times \frac{3}{x-1} \) by cancelling the \( x-1 \) term, resulting in \( (x-1) \times 3 \). So now we multiply this further by \( \frac{x+4}{3x} \), leading to \( \frac{3(x-1)(x+4)}{3x} \). The 3s cancel out.

Step 3: Further Simplify and Cancel Terms

From the expression \( \frac{3(x-1)(x+4)}{3x} \), cancel the factor 3 in the numerator and denominator, leaving \( \frac{(x-1)(x+4)}{x} \). Expand \( (x-1)(x+4) \) to get \( x^2 + 4x - x - 4 \), simplifying to \( x^2 + 3x - 4 \). So the expression becomes \( \frac{x^2 + 3x - 4}{x} \).

Step 4: Identify Restrictions on Variables

The original or simplified expression is not defined if the denominator equals zero. In the expression \( \frac{(x-1)(x+4)}{x} \), the denominator is zero when \( x = 0 \). In general, when \( \frac{x-1}{3} \) is involved in division, \( x \) cannot be 1. Therefore, the expression is undefined for \( x = 0 \) and \( x = 1 \).

Key Concepts

Factoring Polynomials

When faced with expressions like \(x^2 - 2x + 1\), one crucial step in simplifying is to factor polynomials. This expression is what we call a perfect square trinomial. Perfect square trinomials take the form \(a^2 - 2ab + b^2\) and can be factored into \((a - b)^2\).
To factor \(x^2 - 2x + 1\), recognize it as \((x - 1)^2\) since:

• \((x)^2 = x^2\)
• \(2(x)(1) = -2x\)
• \((1)^2 = 1\)

Factoring helps simplify mathematical expressions, making further calculations easier. This process transforms the original quadratic expression into a product of binomials, setting the stage for further simplification in subsequent steps.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. After factoring, it's particularly handy in operations like multiplication or division of fractions.
Consider the simplified multiplication step: \((x - 1)^2 \times \frac{3}{x - 1}\). Here, notice the common term \((x - 1)\). By canceling out one \((x - 1)\) from the numerator and the denominator, you simplify the expression to \((x - 1) \times 3\). Continuing, multiply by another fraction \(\frac{x+4}{3x}\). Here, the 3s cancel out again:

• Numerator: \((x-1)(x+4)\)
• Denominator: \(x\)

The result reduces further when expanded or simplified: turns into \(\frac{x^2 + 3x - 4}{x}\), easily done step by step. Canceling common factors and simplifying makes the fraction clearer and neatly conveys relationships among terms.

Variable Restrictions

Understanding variable restrictions ensures your expressions remain valid. These restrictions help identify when a fraction is not defined, typically when the denominator equals zero.
In the problem, the denominator reaches zero at specific values:

• With \(\frac{(x-1)(x+4)}{x}\), \(x\) cannot be 0, because dividing by zero makes the expression undefined.
• Initially, the division \(\frac{x-1}{3}\) sets another restriction that \(x\) cannot be 1, or the division by zero occurs.

It's crucial to account for these restrictions. Identifying these values prevents undefined mathematical operations and ensures accuracy and validity in results. Remember, always address restrictions to maintain sound expressions when working with rational expressions.