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Chapter 0: Problem 71

Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-9 x-18$$

Short Answer

Expert verified
The completely factored form of the polynomial \(x^{3}+2 x^{2}-9 x-18\) is \((x+2)(x-3)(x+3)\).

Step by step solution

01

Identify and verify the polynomial

The given polynomial is \(x^{3}+2 x^{2}-9 x-18\). Verify whether the polynomial can be factored.
02

Factor by grouping

Group the polynomial in pairs as \((x^{3}+2 x^{2})- (9 x+18)\)
03

Factor out the Greatest common factor (GCF)

In the first pair, factor out \(x^{2}\) to get \(x^{2}(x+2)\). In the second pair, factor out 9 to get \(9(x+2)\). Combine the factored pairs as \(x^{2}(x+2) - 9(x+2)\).
04

Group again

Both the terms have a common factor of \((x+2)\). Factor out \((x+2)\) to get \((x+2)(x^{2}-9)\)
05

Continue to factor until prime

\((x^{2}-9)\) is a difference of squares, which can be further factored to \((x-3)(x+3)\), so the complete factoring of the polynomial is \((x+2)(x-3)(x+3)\).

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