Write each rational expression in lowest terms. x³ + 64 / x + 4

Verified step by step guidance

1. Recognize that the numerator \(x^3 + 64\) is a sum of cubes, which can be factored using the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 4\) because \$64 = 4^3$.
2. Apply the sum of cubes factorization to the numerator: \(x^3 + 64 = (x + 4)(x^2 - 4x + 16)\).
3. Rewrite the original expression by substituting the factored form of the numerator: \(\frac{x^3 + 64}{x + 4} = \frac{(x + 4)(x^2 - 4x + 16)}{x + 4}\).
4. Since \(x + 4\) appears in both the numerator and denominator, and assuming \(x \neq -4\) to avoid division by zero, cancel out the common factor \(x + 4\).
5. The expression in lowest terms is then \(x^2 - 4x + 16\).

Key Concepts

Factoring Sum of Cubes

The expression x^3 + 64 is a sum of cubes since 64 = 4^3. It can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here, a = x and b = 4, so factoring helps simplify the rational expression.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring the numerator and denominator and then canceling common factors. This process reduces the expression to its lowest terms, making it easier to work with or interpret.

Polynomial Division and Cancellation

After factoring, if the denominator is a factor of the numerator, it can be canceled out. Understanding how polynomial terms divide and cancel is essential to correctly simplify the expression without changing its value.