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

Verified step by step guidance

1. Factor the numerator $8m^2+6m-9$ by looking for two numbers that multiply to $8\times -9=-72$ and add to $6$.
2. Rewrite the middle term $6m$ using the two numbers found in the previous step, and factor by grouping.
3. Factor the denominator $16m^2-9$ as a difference of squares: $(4m{)}^2-3^2$.
4. Express the denominator as $(4m-3)(4m+3)$ using the difference of squares formula.
5. Simplify the rational expression by canceling out any common factors in the numerator and the denominator.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To simplify a rational expression, one must factor both the numerator and the denominator to identify and cancel any common factors. Understanding how to manipulate polynomials is essential for working with rational expressions.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is crucial for simplifying rational expressions, as it allows for the identification of common factors in the numerator and denominator. Techniques such as grouping, using the quadratic formula, or recognizing special products are commonly employed in factoring.

Lowest Terms

A rational expression is said to be in lowest terms when the numerator and denominator have no common factors other than 1. To achieve this, one must fully factor both parts and cancel any common factors. This concept is important for ensuring that the expression is simplified to its most basic form, making it easier to work with in further calculations.