Write each rational expression in lowest terms. See Example 2. (8m² + 6m - 9) / (16m² - 9)

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational expression separately. The expression is \( \frac{8m^{2} + 6m - 9}{16m^{2} - 9} \).
2. Factor the numerator \(8m^{2} + 6m - 9\). Look for two numbers that multiply to \(8 \times (-9) = -72\) and add to \(6\). Use these to split the middle term and factor by grouping.
3. Factor the denominator \(16m^{2} - 9\). Recognize this as a difference of squares, which factors as \(a^{2} - b^{2} = (a - b)(a + b)\).
4. After factoring numerator and denominator, write the expression as a product of factors over a product of factors.
5. Cancel out any common factors that appear in both numerator and denominator to write the expression in lowest terms.

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of simpler binomials or polynomials. For example, to simplify rational expressions, you factor both numerator and denominator to identify common factors. Recognizing patterns like the difference of squares or trinomial factoring is essential.

Difference of Squares

The difference of squares is a special factoring pattern where an expression of the form a² - b² factors into (a - b)(a + b). This is useful for simplifying denominators or numerators that fit this pattern, allowing cancellation of common factors in rational expressions.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their lowest terms by factoring numerator and denominator and canceling common factors. This process makes expressions easier to work with and understand, and it is crucial for solving equations or performing further operations.