What is the factorization of the expression x^3 + 2x^2 - 9x - 18?

To factor the expression ( x^3 + 2x^2 - 9x - 18 ), we can use the method of grouping or synthetic division to find its roots.

First, we can look for rational roots using techniques such as the Rational Root Theorem, which suggests that any potential rational root is a factor of the constant term (-18) over a factor of the leading coefficient (1). Testing some possible roots, we find that ( x = -2 ) is a root because substituting it into the equation yields zero.

Now that we know ( x + 2 ) is a factor, we can divide the original polynomial by ( x + 2 ) using synthetic division or polynomial long division. Doing so gives us a quotient of ( x^2 - 9 ).

Next, we can factor ( x^2 - 9 ) further as a difference of squares, which becomes ( (x - 3)(x + 3) ).

Thus, putting everything together, the complete factorization of ( x^3 + 2x^2 - 9x - 18 ) can be expressed as:

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(x + 2)(x - 3)(x