Write each rational expression in lowest terms. 8m² + 6m - 9 / 16m² - 9

Verified step by step guidance

1. Identify the rational expression given: \(\frac{8m^2 + 6m - 9}{16m^2 - 9}\).
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\(. Recognize this as a difference of squares and apply the formula \)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 any common factors that appear in both numerator and denominator to write the rational expression in lowest terms.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This is essential for simplifying rational expressions by identifying common factors in the numerator and denominator. Techniques include factoring trinomials, difference of squares, and grouping.

Rational Expressions

A rational expression is a fraction where the numerator and denominator are polynomials. Simplifying rational expressions requires factoring both parts and canceling common factors, similar to simplifying numerical fractions.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their lowest terms by canceling common polynomial factors. This process makes expressions easier to work with and is crucial for solving equations or performing operations involving rational expressions.