Multiply or divide as indicated. [(x^2+6x+9)(x+3)]/[(x^2-4)(x-2)]

검증된 단계별 안내

Factorize the numerator and denominator completely. For the numerator, factor \(x^2 + 6x + 9\) as \((x+3)(x+3)\), since it is a perfect square trinomial. The denominator \(x^2 - 4\) is a difference of squares, so factor it as \((x+2)(x-2)\).

Rewrite the expression using the factored forms: \[\frac{(x+3)(x+3)(x+3)}{(x+2)(x-2)(x-2)}\].

Simplify the expression by canceling out any common factors in the numerator and denominator. In this case, there are no common factors to cancel between the numerator and denominator.

Combine the remaining terms in the numerator and denominator, if possible, to simplify the expression further. The numerator becomes \((x+3)^3\), and the denominator becomes \((x+2)(x-2)^2\).

Write the final simplified expression as \[\frac{(x+3)^3}{(x+2)(x-2)^2}\]. Ensure that any restrictions on the variable are noted, such as \(x \neq -2, 2\), to avoid division by zero.

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주요 개념

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

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. For example, the expression x^2 + 6x + 9 can be factored into (x + 3)(x + 3) or (x + 3)^2. This process is essential for simplifying expressions and solving equations, particularly when dealing with rational expressions.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including multiplying, dividing, and simplifying them, is crucial in algebra. In the given question, the rational expression involves both multiplication and division of polynomials, requiring careful handling of each component.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes canceling common factors in the numerator and denominator. This is particularly important in rational expressions, as it can make calculations easier and clearer. In the context of the question, simplifying the expression after multiplication and division will yield a more manageable result.