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

Verified step by step guidance

1. Rewrite the equation in standard form by moving all terms to one side of the equation: \(x^3 + 2x^2 - 9x - 18 = 0\).
2. Factor the equation by grouping. Group the terms as \((x^3 + 2x^2)\) and \((-9x - 18)\).
3. Factor out the greatest common factor (GCF) from each group: \(x^2(x + 2) - 9(x + 2) = 0\).
4. Notice that \((x + 2)\) is a common factor. Factor it out: \((x + 2)(x^2 - 9) = 0\).
5. 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 \(x + 3 = 0\).

Key Concepts

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.