Problem 79 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The factorized form of the polynomial \(x^{3}+2 x^{2}-4 x-8\) is \((x+2)^{2}(x-2)\).

Step 1 Identify the polynomial

The polynomial provided in the exercise is \(x^{3}+2 x^{2}-4 x-8\).

Step 2 Factor out the polynomial

Factorizing involves breaking the polynomial down into its most basic factors. In this case, we don't have any common factor in all terms, therefore, we need another approach. In this case, we can try grouping which is suitable for a four-term polynomial. This requires splitting the polynomial into two groups, then factor out a common polynomial from each group. However, as seen there is no grouping that can provide us with a common factor.

Step 3 Synthetic Division

To factor further, we consider synthetic division method. First we need to select a root of the polynomial to use for synthetic division. We usually start by testing the factors of the constant term, that is \(\pm1\), \(\pm2\), \(\pm4\), and \(\pm8\). If we test these in the polynomial, we find that the root that will make the polynomial equal to zero is -2. Thus conduct the synthetic division of \(x^{3}+2 x^{2}-4 x-8\) by \((x+2)\) to obtain a quadratic function.

Step 4 Factoring the Quadratic Expression

After synthetic division, we get a quadratic expression which is \(x^{2}-4\). To factorize this, we use the differences of squares factoring rule where \(a^{2} - b^{2} = (a-b)(a+b)\). Thus, \(x^{2}-4\) will be \((x-2)(x+2)\).

Step 5 Final Factored Form

Now, combine \((x+2)\) obtained from synthetic division with \((x-2)(x+2)\) factored from the quadratic expression to form the completely factored polynomial which is \((x+2)^{2}(x-2)\).