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Chapter 0: Problem 91

Factor completely, or state that the polynomial is prime. $$2 x^{3}-8 a^{2} x+24 x^{2}+72 x$$

Short Answer

Expert verified
The completely factored form of the polynomial \(2x^{3}-8a^{2}x+24x^{2}+72x\) is \(2x^{2}(x + 12) - 8x(a^{2} - 9)\).

Step by step solution

01

Order the terms

Firstly, arrange the terms of the polynomial according to the descending power of the variable. So, \(2x^{3}+24x^{2}-8a^{2}x+72x\).
02

Group Common terms

Group the terms that have common factors. Grouping the terms will make the job of factoring easier. Hence, \(2x^{3}+24x^{2} - 8a^{2}x + 72x\) can be written as \((2x^{3}+24x^{2}) - (8a^{2}x - 72x)\).
03

Factoring out common terms

The next step is to factor out the common terms from each group. Factoring out means to find the common factor of the terms and write it outside the brackets, allowing the terms inside the brackets to be reduced. From \(2x^{3}+24x^{2}\), pull out \(2x^{2}\) to get \(2x^{2}(x + 12)\). And from \(8a^{2}x - 72x\), pull out \(8x\) to get \(8x(a^{2} - 9)\). Hence, the final factored polynomial is \(2x^{2}(x + 12) - 8x(a^{2} - 9)\).

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