Problem 25 - Precalculus : Building Concepts and Connections

Question

Solve the polynomial inequality. $$x^{3}>2 x^{2}+3 x$$

Step-by-Step Solution

Answer

The solution is \((-∞, -3) \cup (0, \infty)\).

Step 1: Rearrange polynomial to one side of inequality

Rearrange the given inequality \(x^{3} > 2 x^{2} + 3 x\). Set all terms to one side of the inequality, resulting in \(x^{3} - 2 x^{2} - 3x > 0\)..

Step 2: Factor the polynomial

Factor the polynomial on the left side of the inequality. \(x(x^{2} - 2x - 3) > 0\). Further factoring the quadratic inside parentheses gives \(x(x - 3)(x + 1) > 0\)

Step 3: Use a number line to find the solution

To find where the polynomial is greater than 0, mark possible roots on the number line -3, 0, and 3. Plug in different values between these roots into the original equation to see if the inequality is satisfied. For example, choose -4, -1, 1, 4 as test points. With these test points plugged into our factored inequality, -4 and 1 make the inequality true, so these regions are included in the solution set. So, the solution set is \((-∞, -3) \cup (0, \infty)\).

Step 4: Write Final Answer

The final solution can be described as the union of two intervals: negative infinity to -3 and 0 to positive infinity.

Key Concepts

Factoring Polynomials

When solving polynomial inequalities, an essential step is factoring the polynomial. Factoring involves expressing the polynomial as a product of its simplest components. These simpler forms make it easier to analyze where the polynomial is positive or negative.
To factor a polynomial like \(x^3 - 2x^2 - 3x\), we look for common factors among the terms. Here, all the terms share \(x\) as a factor, leading to \(x(x^2 - 2x - 3)\). We then focus on factoring the quadratic \(x^2 - 2x - 3\).
The quadratic can further be factored by looking for two numbers that multiply to -3 and add to -2, resulting in \((x-3)(x+1)\). Hence, the complete factorization of the polynomial is \(x(x-3)(x+1)\). This helps in determining where the inequality \(x(x-3)(x+1) > 0\) holds true.

Inequalities

Inequalities are mathematical expressions involving the symbols \(<\), \(>\), \(\leq\), or \(\geq\). They represent a range of values that satisfy a given condition. In dealing with polynomial inequalities, the goal is to find intervals where the polynomial is either positive or negative.
Consider the inequality \(x(x-3)(x+1) > 0\). This requires us to determine the values of \(x\) at which the expression is greater than zero. Inequalities are generally solved by identifying critical points, or roots, which in this example are \(-3, 0,\) and \(3\).
Once the roots are identified, we can use a number line to test intervals between these points to ascertain where the polynomial expression maintains the inequality. By evaluating test points in each interval, we decide which sections of the number line satisfy our condition \(x(x-3)(x+1) > 0\).

Solution Sets

When solving inequalities, the solution set comprises all possible values of \(x\) that satisfy the inequality. After identifying and testing intervals separated by critical points, the solution set is presented using interval notation.
Using the example polynomial \(x(x-3)(x+1) > 0\), once calculations show the intervals where the expression is positive, the solution set is defined. Intervals where test points meet the original inequality are included in the solution set.
Here, the test points reveal that the polynomial is positive in the intervals \((-\infty, -3)\) and \((0, \infty)\). This is combined as the union of these intervals: \((-\infty, -3) \cup (0, \infty)\). This signifies the complete range of solutions that satisfy the inequality.