Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18

검증된 단계별 안내

Rewrite the equation in standard form by moving all terms to one side of the equation: \(x^3 + 2x^2 - 9x - 18 = 0\).

Factor the equation by grouping. Group the terms as \((x^3 + 2x^2)\) and \((-9x - 18)\).

Factor out the greatest common factor (GCF) from each group: \(x^2(x + 2) - 9(x + 2) = 0\).

Notice that \((x + 2)\) is a common factor. Factor it out: \((x + 2)(x^2 - 9) = 0\).

Factor \(x^2 - 9\) further using the difference of squares: \((x + 2)(x - 3)(x + 3) = 0\). Solve for \(x\) by setting each factor equal to zero: \(x + 2 = 0\), \(x - 3 = 0\), and \(x + 3 = 0\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Polynomial Equations

A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms consisting of variables raised to non-negative integer powers. In this case, the equation x^3 + 2x^2 - 9x - 18 = 0 is a cubic polynomial equation. Understanding how to manipulate and solve polynomial equations is essential for finding the roots of the equation.

Factoring

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This method is often used to solve polynomial equations by setting each factor equal to zero. In the given equation, factoring can simplify the process of finding the values of x that satisfy the equation.

The Rational Root Theorem

The Rational Root Theorem provides a way to identify possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem can guide the search for potential solutions to the equation x^3 + 2x^2 - 9x - 18 = 0.