Multiply or divide as indicated. $\frac{x^3 - 8}{x^2 - 4} \cdot \frac{x + 2}{3x}$

Verified step by step guidance

1. Identify the given expression to multiply: \(\frac{\left(x^3 - 8\right)}{\left(x^2 - 4\right)} \cdot \frac{\left(x + 2\right)}{3x}\).
2. Factor all polynomials where possible. Recognize that \(x^3 - 8\) is a difference of cubes and \(x^2 - 4\) is a difference of squares. Use the formulas: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) and \(a^2 - b^2 = (a - b)(a + b)\).
3. Rewrite the expression with factored forms: \(\frac{(x - 2)(x^2 + 2x + 4)}{(x - 2)(x + 2)} \cdot \frac{(x + 2)}{3x}\).
4. Cancel out common factors in the numerator and denominator, such as \((x - 2)\) and \((x + 2)\), to simplify the expression.
5. Multiply the remaining factors in the numerator and denominator to write the simplified expression as a single fraction.

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting expressions as products of simpler polynomials. Recognizing special forms like difference of cubes (x³ - 8) and difference of squares (x² - 4) helps simplify expressions before multiplication or division.

Multiplication and Division of Rational Expressions

When multiplying or dividing rational expressions, factor all numerators and denominators first, then multiply across numerators and denominators. For division, multiply by the reciprocal of the divisor to simplify the expression.

Simplifying Rational Expressions

Simplifying involves canceling common factors in the numerator and denominator after factoring. This reduces the expression to its simplest form, making it easier to interpret or use in further calculations.