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

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

All textbooks

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 31a

Chapter 1, Problem 31a

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

Verified step by step guidance

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 $64 = 4^3$.

Apply the sum of cubes factorization to the numerator: $x^3 + 64 = (x + 4)(x^2 - 4x + 16)$.

Rewrite the original expression by substituting the factored form of the numerator: $\frac{x^3 + 64}{x + 4} = \frac{(x + 4)(x^2 - 4x + 16)}{x + 4}$.

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 $x + 4$.

The expression in lowest terms is then $x^2 - 4x + 16$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

Guided course

05:13

Introduction to Polynomials

Related Practice

Textbook Question

Insert ∈ or ∉ in each blank to make the resulting statement true. 0 _____ {0, 5, 6, 7, 8, 10}

Textbook Question

Insert ∈ or ∉ in each blank to make the resulting statement true. 0 _____ {0, 2, 3, 4}

Textbook Question

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. ⁵√ k²

views

Textbook Question

Find each product. $x^{2} (3 x - 2) (5 x + 1)$

views

Textbook Question

Simplify each expression. Assume all variables represent nonzero real numbers. (-5n⁴/r²)³

Textbook Question

Multiply or divide as indicated. Write answers in lowest terms as needed. (21/8)∙(4/7)