Write each rational expression in lowest terms. See Example 2. (8x² + 16x) / 4x²

Verified step by step guidance

1. Start by writing the given rational expression as a fraction: \(\frac{8x^{2} + 16x}{4x^{2}}\).
2. Factor the numerator by taking out the greatest common factor (GCF). For \$8x^{2} + 16x\(, the GCF is \)8x\(, so rewrite the numerator as \)8x(x + 2)$.
3. Rewrite the expression with the factored numerator: \(\frac{8x(x + 2)}{4x^{2}}\).
4. Factor the denominator if possible. Here, \$4x^{2}$ is already factored as \(4 \cdot x^{2}\).
5. Simplify the fraction by dividing both numerator and denominator by their common factors. Identify and cancel out the common factors step-by-step.

Key Concepts

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of its factors. For example, 8x² + 16x can be factored by taking out the greatest common factor (GCF), which is 8x, resulting in 8x(x + 2). Factoring simplifies expressions and is essential for reducing rational expressions.

Simplifying Rational Expressions

A rational expression is a fraction where the numerator and denominator are polynomials. Simplifying involves factoring both parts and canceling common factors. This process reduces the expression to its lowest terms, making it easier to work with or interpret.

Greatest Common Factor (GCF)

The GCF is the largest factor shared by two or more terms or polynomials. Identifying the GCF helps in factoring expressions and simplifying rational expressions by canceling common factors in numerator and denominator.