Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Find all real solutions of the polynomial equation. $$x^{4}-13 x^{2}-12 x=0$$

Step-by-Step Solution

Answer

The real solutions of the polynomial equation are x=0, x=4, x=-3 and x=1.

Step 1: Factoring Out a Common Variable

First, we can notice that x is a common factor. Therefore we factor it out, obtaining \(x(x^{3}-13x-12) = 0\).

Step 2: Factoring the Quadratic Equation

Next, we need to factor the quadratic equation \(x^{3}-13x-12\). To do this, we try to rewrite as \((x-a)(x-b)(x-c) = 0\). By trying several combinations we find that a=4, b=-3 and c=1 work, hence our factored polynomial equation becomes \(x(x-4)(x+3)(x-1) = 0.\)

Step 3: Solving for Each Factor

Setting each factor to zero results in the following solutions: \(x = 0\), \(x = 4\), \(x = -3\) and \(x = 1\). These are the values for x that make the polynomial equation true.