In Exercises 15–32, multiply or divide as indicated. (6x+9)/(3x−15) ⋅ (x−5)/(4x+6)

Verified step by step guidance

1. Factor the numerators and denominators where possible. For the first fraction, factor out a 3 from the numerator and factor the denominator as a difference of squares. For the second fraction, factor out a 2 from the denominator.
2. Simplify the expressions by canceling out common factors in the numerator and denominator across the fractions.
3. Multiply the simplified numerators together and the simplified denominators together to form a new fraction.
4. Check if the resulting fraction can be simplified further by factoring and reducing any common factors.
5. Write the final expression in its simplest form.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, multiplying, and dividing them, is crucial for solving problems involving them. In this case, the expression (6x+9)/(3x−15) is a rational expression that can be simplified before performing operations.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, as it allows for cancellation of common factors in the numerator and denominator. For example, both the numerator and denominator in the given expression can be factored to reveal simpler forms that facilitate multiplication and division.

Multiplication of Rational Expressions

When multiplying rational expressions, the process involves multiplying the numerators together and the denominators together. It is important to simplify the resulting expression by canceling any common factors before finalizing the answer. This method ensures that the multiplication is performed correctly and efficiently, leading to a simplified result.