Find the domain of each rational expression. x 3 - 1 / x - 1

Hello Everyone. We're going to solve for the domain of X to the 3rd -400 over X -20. So rewriting that so I can clearly see my numerator by denominator. I again have X to the 3rd -400 over X -20. The domain is all the X values that apply without making this be an undefined fraction. If you recall an undefined fraction happens when the denominator equal zero. So in this case we want to find out what would make the denominator equal zero. So we're going to set the denominator equal to zero. So we have X minus 20 equals zero solving for that. We find that The denominator will be zero when x equals 20. What this means is that our domain is all real numbers Except when X is 20. So all real numbers except 20. What that looks like an interval notation, which is what our solutions are set up like is that we go from negative infinity All the way to 20, But 20 is not included. And then we jump back into 20 all the way to positive infinity. So that is my solution. And that is answer choice B. Have a good day

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Problem 19a

Textbook Question

Find the domain of each rational expression. x³ - 1 / x - 1

Verified step by step guidance

Identify the rational expression given: \(\frac{x^3 - 1}{x - 1}\).

Recall that the domain of a rational expression excludes values that make the denominator zero, because division by zero is undefined.

Set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\).

Solve the equation to find the value(s) to exclude from the domain: \(x = 1\).

Conclude that the domain is all real numbers except \(x = 1\), which can be written in interval notation as \((-\infty, 1) \cup (1, \infty)\).

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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 denominator are polynomials. Understanding rational expressions involves knowing how to simplify, factor, and analyze these fractions, especially focusing on restrictions caused by the denominator.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational expressions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This skill helps identify values that make the denominator zero by factoring and setting each factor equal to zero to find restrictions on the domain.

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