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

Verified step by step guidance

1. Factor the numerator: $8x^2+16x=8x(x+2)$.
2. Factor the denominator: $4x^2=4x\cdot x$.
3. Identify the common factors in the numerator and the denominator.
4. Cancel out the common factors from the numerator and the denominator.
5. Write the simplified expression after canceling the common factors.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, as it allows us to identify common factors in the numerator and denominator. For example, in the expression 8x² + 16x, we can factor out 8x, resulting in 8x(x + 2).

Lowest Terms

A rational expression is in lowest terms when the numerator and denominator have no common factors other than 1. To simplify a rational expression, we divide both the numerator and denominator by their greatest common factor (GCF). This process ensures that the expression is as simple as possible, making it easier to work with in further calculations.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial in algebra and calculus. In the given problem, we are tasked with simplifying a rational expression, which requires knowledge of polynomial operations and simplification techniques.