Write each rational expression in lowest terms. x^3 + 64 / x + 4

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To simplify a rational expression, one must factor both the numerator and the denominator and then cancel any common factors. Understanding how to manipulate these expressions is crucial for solving algebraic problems.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For example, the expression x^3 + 64 can be factored using the sum of cubes formula, a^3 + b^3 = (a + b)(a^2 - ab + b^2). Recognizing and applying factoring techniques is essential for simplifying rational expressions.

Lowest Terms

A rational expression is in lowest terms when the numerator and denominator have no common factors other than 1. To achieve this, one must fully factor both parts and eliminate any shared factors. This process ensures that the expression is simplified to its most basic form, making it easier to work with in further calculations.