Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

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Factor the numerator \(x^3 + 2x^2\). Start by factoring out the greatest common factor (GCF), which is \(x^2\). This gives \(x^2(x + 2)\).

Identify the denominator \(x + 2\). Note that the denominator cannot be zero, so exclude \(x = -2\) from the domain.

Simplify the rational expression by canceling out the common factor \(x + 2\) in the numerator and denominator. This is valid as long as \(x \neq -2\).

After canceling, the simplified expression is \(x^2\).

Summarize the solution: The simplified expression is \(x^2\), and the excluded value from the domain is \(x = -2\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions involves factoring both parts and reducing them by canceling out common factors. Understanding how to manipulate polynomials is essential for simplifying rational expressions effectively.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that when multiplied together give the original polynomial. This is crucial in simplifying rational expressions, as it allows for the identification and cancellation of common factors in the numerator and denominator.

Domain of a Rational Expression

The domain of a rational expression consists of all the possible values of the variable that do not make the denominator equal to zero. Identifying these values is important because they must be excluded from the domain to avoid undefined expressions. In the given example, finding the values that make the denominator zero is essential for determining the domain.