Textbook Question
Write each rational expression in lowest terms. 3(3 - t) / (t + 5)(t - 3)
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0. Review of Algebra
Factoring Polynomials
2:41 minutesProblem 31
Textbook Question
Write each rational expression in lowest terms. x^3 + 64 / x + 4
Verified step by step guidance1
Identify the expression: \( \frac{x^3 + 64}{x + 4} \). This is a rational expression where the numerator is a sum of cubes.
Recognize that \( x^3 + 64 \) 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 \( b = 4 \).
Apply the sum of cubes formula: \( x^3 + 64 = (x + 4)(x^2 - 4x + 16) \).
Substitute the factored form into the original expression: \( \frac{(x + 4)(x^2 - 4x + 16)}{x + 4} \).
Cancel the common factor \( x + 4 \) from the numerator and the denominator, resulting in the simplified expression \( x^2 - 4x + 16 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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