Problem 42 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3} \leq 4 x^{2} $$

Step-by-Step Solution

Answer

The solution to the inequality \(x^{3} \leq 4x^{2}\) is \(x \in [- \infty, 4]\)

Step 1: Rearrange the inequality

The first step is to rearrange the inequality. Move \(4x^{2}\) from the right hand side of the inequality to the left hand side by subtracting \(4x^{2}\) from both sides. This will produce a new, equivalent inequality: \(x^{3} - 4x^{2} \leq 0\).

Step 2: Factor the polynomial

The next step is factoring the polynomial. The polynomial \(x^{3} - 4x^{2} \leq 0\) can be factored by taking out the greatest common factor, which is \(x^{2}\). After factoring, we get the inequality \(x^{2}(x - 4) \leq 0\).

Step 3: Find the roots

The next step is to find the roots of the polynomial \(x^{2}(x - 4)\). The roots are the solutions to the equation \(x^{2}(x - 4) = 0\). From this equation, it's clear that the roots are \(x = 0\) and \(x = 4\).

Step 4: Use the roots to create the intervals

Using the roots that found from the previous step, three intervals can be created on the real number line. These intervals are \(-\infty < x < 0\), \(0 < x < 4\), and \(4 < x < \infty \).

Step 5: Determine the signs in the intervals

To determine the sign of the polynomial in each interval, use an x-value from each interval, plug these x-values back into the polynomial inequality and check the sign. For \(-\infty < x < 0\), use \(x=-1\), and find \((-1)^2((-1) - 4) = -5\), which is less than 0. For \(0 < x < 4\), use \(x=2\), and find \(2^2(2 - 4) = -8\), which is less than 0. For \(4 < x < \infty \), use \(x=5\), and find \(5^2(5 - 4) = 25\), which is more than 0. So, the sign for the first given interval is -, for the second interval is - and for the third interval is +.

Step 6: Write the Solution in Interval Notation

The polynomial is less than or equal to zero when \(x\) belongs to \([- \infty, 0]\) and \([0, 4]\). The union of these two intervals gives the solution to the inequality: \(x\) belongs to \([- \infty, 4]\)