Problem 91 - College Algebra Essentials

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 completely factored form of the given polynomial is \(2x[(x - 2a + 6)^2]\)

Step 1: Identify the common factor

To factor the polynomial, first we need to identify the greatest common factor (GCF) of all the terms. In this case, the GCF of all terms is \(2x\).

Step 2: Factor out the common factor

Factor out the common factor, which gives: \(2x(x^2 - 4a^2 + 12x + 36)\)

Step 3: Further factoring

Next, it can be identified that the expression inside the brackets is a quadratic equation and can be factored. This leads to \(2x[(x - 2a)^2 + 12(x - 2a) + 36]\)

Step 4: Complete the square

The expression inside the brackets is a perfect square trinomial and can be further written as a square of a binomial. Completing the square we get: \(2x[(x - 2a + 6)^2]\)