Problem 91 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polynomial \(2x^{3}-8a^{2}x+24x^{2}+72x\) is not prime and can be factored completely as \(x [(\sqrt{2x}+2a)(\sqrt{2x}-2a) + 24(x+3) ]\)

Step 1: Identify Common Factors

The given polynomial is \(2x^{3}-8a^{2}x+24x^{2}+72x\). The first step is to check for common factors. All terms have x as a common factor, so we can factor it out: \(x(2x^{2}-8a^{2}+24x+72)\)

Step 2: Factor Quadratic Term

Now we'll separate the quadratic term \(2x^{2}-8a^{2}\) and the rest, giving us \(x(2x^{2}-8a^{2}+24x+72) = x[(2x^{2}-8a^{2})+24x+72]\). The term \(2x^{2}-8a^{2}\) is a difference of two squares, which can be factored as \((\sqrt{2x})^{2}- (2a)^2= (\sqrt{2x}+2a)(\sqrt{2x}-2a)\)

Step 3: Factor Linear and Constant term

Consider the linear term and the constant term, 24x+72. This can be factored since 24 and 72 have a common factor of 24, giving us: 24(x+3).

Step 4: Combine All Parts

Now, we combine all the factors from steps 2 and 3: \(x [(\sqrt{2x}+2a)(\sqrt{2x}-2a) + 24(x+3) ]\)

Step 5: Simplify the polynomial

It was observed that the polynomial is not prime, and has been successfully factored as \(x [(\sqrt{2x}+2a)(\sqrt{2x}-2a) + 24(x+3) ]\)