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Chapter 0: Problem 79

Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-4 x-8$$

Short Answer

Expert verified
Therefore, \(x^{3} +2x^{2} -4x -8\) can be factored completely as \((x -2)(x +2)^2\).

Step by step solution

01

Group Terms

In order to factor the polynomial \(x^{3} +2x^{2} -4x -8\), first group the terms in a way that a common factor can be taken out. So, the polynomial can be written as: \((x^{3} +2x^{2}) + (-4x -8)\)
02

Factor out Common Terms

In each group, take out the common factors. So, this leads to: \(x^{2}(x +2) -4(x +2)\)
03

Factor out Common Binomial

Now, we notice that both terms have a common binomial \(x +2\). So, factor out the binomial \(x +2\). Thus, the polynomial is factorized as: \((x^{2} -4)(x +2)\)
04

Factorize further

\(x^{2} -4\) is a difference of squares and can be factorized further as \((x -2)(x +2)\). Therefore, the given polynomial can be factorized completely as: \((x -2)(x +2)(x +2)\)

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