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Chapter 0: Problem 76

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

Short Answer

Expert verified
So, the completely factorized form of the polynomial \(9 x^{3}-9 x\) is \(9x(x-1)(x+1)\).

Step by step solution

01

Identifying the Common Factor

The first step in factorizing the polynomial \(9 x^{3}-9 x\) completely is to identify the common factor. In this expression, \(9x\) is the common factor.
02

Applying Distributive Property

Next step is to apply the distributive property to factor out the common factor. The distributive property is \(a(b + c) = ab + ac\). Here we will be using the reverse of the distributive property, factoring out common factor from each term: \(9x( x^{2}-1)\).
03

Apply the Difference of Squares

The third step is to further factorize \(x^{2}-1\). This term is a difference of squares, which can be factorized as \((a+b)(a-b)\) where \(a=x\) and \(b=1\). The expression becomes: \(9x(x-1)(x+1)\).

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