Write each rational expression in lowest terms. 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.
2. Factor the numerator \(m^2 - 4m + 4\). Look for two numbers that multiply to 4 and add to -4.
3. Factor the denominator \(m^2 + m - 6\). Find two numbers that multiply to -6 and add to 1.
4. Rewrite the rational expression using the factored forms of the numerator and denominator.
5. Simplify the expression by canceling out any common factors that appear in both the numerator and denominator.

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This process helps simplify rational expressions by identifying common factors in the numerator and denominator. For example, m^2 - 4m + 4 factors to (m - 2)(m - 2).

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their lowest terms by canceling common factors in the numerator and denominator. After factoring, any shared binomial factors can be divided out, making the expression simpler and easier to work with.

Identifying Restrictions on the Variable

When simplifying rational expressions, it's important to determine values of the variable that make the denominator zero, as these are excluded from the domain. For m^2 + m - 6, factoring helps find these values, ensuring the simplified expression is valid only where the original is defined.