Write each rational expression in lowest terms. 8x^2 + 16 / 4x^2

Verified step by step guidance

1. Identify the numerator and the denominator of the rational expression. The numerator is \(8x^2 + 16\) and the denominator is \(4x^2\).
2. Factor the numerator \(8x^2 + 16\). Notice that both terms have a common factor of 8, so factor out 8 to get \(8(x^2 + 2)\).
3. Observe the denominator \(4x^2\). It is already factored as \(4 \cdot x^2\).
4. Simplify the expression by canceling out the common factors in the numerator and the denominator. The common factor here is 4.
5. Rewrite the expression with the remaining factors after cancellation to express the rational expression in its lowest terms.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To work with rational expressions, it is essential to understand how to manipulate polynomials, including addition, subtraction, multiplication, and division. Simplifying these expressions often involves factoring and reducing them to their lowest terms.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This is crucial for simplifying rational expressions, as it allows for the cancellation of common factors in the numerator and denominator, leading to a simpler form of the expression.

Lowest Terms

An expression is in lowest terms when there are no common factors between the numerator and the denominator other than 1. To achieve this, one must factor both parts of the rational expression and cancel out any common factors. This ensures that the expression is simplified as much as possible, making it easier to work with and understand.