Write each rational expression in lowest terms. See Example 2. (m² - 4m + 4) / (m² + m - 6)

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational expression separately. For the numerator \(m^{2} - 4m + 4\), look for two numbers that multiply to \(4\) and add to \(-4\).
2. Factor the numerator as a perfect square trinomial: \(m^{2} - 4m + 4 = (m - 2)^{2}\).
3. Next, factor the denominator \(m^{2} + m - 6\). Find two numbers that multiply to \(-6\) and add to \(1\).
4. Factor the denominator as \((m + 3)(m - 2)\).
5. Write the rational expression as \(\frac{(m - 2)^{2}}{(m + 3)(m - 2)}\) and then simplify by canceling the common factor \((m - 2)\), leaving the expression in lowest terms.

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For example, m² - 4m + 4 factors to (m - 2)(m - 2). Recognizing patterns like perfect square trinomials or factoring by grouping helps simplify expressions.

Simplifying Rational Expressions

A rational expression is simplified by factoring the numerator and denominator and then canceling common factors. This reduces the expression to its lowest terms, making it easier to work with or evaluate.

Identifying and Excluding Restrictions

When simplifying rational expressions, it's important to identify values that make the denominator zero, as these are excluded from the domain. This ensures the expression remains defined and valid.