Problem 79 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polynomial is not prime and can be factored completely as \((x+2)^2(x-2)\).

Step 1: Find common factors

Observe the polynomial \(x^{3}+2 x^{2}-4 x-8\) and look for any common factors in each term. Here, there are no common factors in all terms, so proceed to the next step.

Step 2: Factor by grouping

Divide the equation into two parts and factor both separately: \((x^{3}+2x^{2}) + (-4x-8)\). The first group can be factored by taking out the common factor of \(x^{2}\), resulting in \(x^{2}(x+2)\). The second group can be factored by taking out the common factor of -4, resulting in -4(x+2). This yields \(x^2(x+2) - 4(x+2)\). Now it can be seen that (x+2) is a common factor.

Step 3: Take out common factor

Take out the common factor (x+2) from both terms, yielding: \((x+2)(x^2-4)\)

Step 4: Factor difference of squares

The second factor, \(x^2-4\), is a difference of squares and can be factored further into \((x-2)(x+2)\). So, the polynomial is factored completely as \((x+2)^2(x-2)\).