Problem 11 - Precalculus : Building Concepts and Connections

Question

Solve the polynomial inequality. $$x^{3}-4 x^{2} \geq 0$$

Step-by-Step Solution

Answer

The solution of the inequality \(x^{3}-4 x^{2} \geq 0\) is \(x \leq 0\) or \(x \geq 4\).

Step 1: Finding the Roots of the Polynomial

Set the polynomial \(x^{3}-4 x^{2} = 0\) and solve for x to find the roots of the polynomial. Factoring out the common terms, we get \(x^{2}(x - 4) = 0\). Thus, the roots of the polynomial are x = 0 and x = 4.

Step 2: Check the intervals defined by the roots

The roots divide the number line into three intervals: \(-\infty4\). To find out where the polynomial is greater than or equal to zero, test a number in each interval: for \(-\infty4\), choose x = 5. Substitute these into the original polynomial inequality \(x^{3}-4 x^{2} \geq 0\). The results are \(-1\) (negative), \(-3\) (negative), and \(5\) (positive), respectively. This implies that the polynomial is less than 0 in the intervals \(-\infty4\).

Step 3: Conclusion

Combine the findings from steps 1 and 2. The polynomial equals zero when x = 0 and x = 4, and is greater than zero when \(x>4\). Therefore, the solution to the polynomial inequality \(x^{3}-4 x^{2} \geq 0\) is \(x \leq 0\) or \(x \geq 4\).

Key Concepts

Factoring Polynomials

Factoring is a fundamental technique in algebra that simplifies polynomials to make solving them easier. In the given exercise, the polynomial is \(x^3 - 4x^2 \). To simplify it, we look for common factors in each term. Here, both terms share \(x^2\) as a factor.
By factoring out \(x^2\), the polynomial becomes \(x^2(x - 4) = 0\). This expression is easier to handle because it shows the factors distinctly. Factoring essentially allows us to express the polynomial in terms of products of simpler expressions. This step is crucial in solving inequalities as it sets the stage for finding roots.
Key steps in factoring polynomials include:

• Identify the greatest common factor (GCF) in all terms.
• Use the distributive property to factor the GCF out.
• Simplify the polynomial into a product of simpler polynomials.

Factoring transforms a potentially complicated problem into a series of simpler ones, making it easier to analyze and solve.

Finding Roots

Finding the roots of the polynomial involves determining the values of \(x\) that make the polynomial equal to zero. In this exercise, after factoring, the polynomial \(x^2(x-4) = 0\) reveals the roots directly.
Setting each factor equal to zero gives us:

• \(x^2 = 0\), which simplifies to \(x = 0\)
• \(x - 4 = 0\), which gives \(x = 4\)

These values are the points where the polynomial switches behavior on the number line. Finding roots is important in solving inequalities because these points serve as boundaries that separate different intervals for testing.
In polynomial inequalities, roots often indicate where the expression changes from positive to negative or vice versa. Understanding these transition points is essential for analyzing the polynomial's behavior over various intervals.

Interval Testing

Interval testing is used after identifying the roots to determine on which intervals the polynomial satisfies the inequality condition. In this case, the polynomial \(x^3 - 4x^2 \geq 0\) has roots at \(x = 0\) and \(x = 4\). These roots create the test intervals:

• \(-\infty < x < 0\)
• \(0 < x < 4\)
• \(x > 4\)

In each interval, we choose a test point (e.g., \(x = -1, 1, 5\)) and substitute it back into the polynomial inequality.
For \(x = -1\) and \(x = 1\), the results are negative, indicating that the polynomial is less than zero on those intervals. At \(x = 5\), the result is positive, showing that the polynomial is greater than zero.Interval testing helps determine the overall behavior of the polynomial across its domain and helps find the solution set that satisfies the inequality, which in this case is \(x \leq 0\) or \(x \geq 4\). This method gives us a clear picture of where the inequality holds true.