Problem 50 - College Algebra with Modeling and Visualization

Question

Solve the polynomial inequality. $$ -(x+1)^{2}(x-2) \geq 0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1] \cup [-1, 2]\).

Step 1: Identify the critical points

First, identify where the polynomial changes sign by finding the roots.The equation given is:\[-(x+1)^2(x-2) = 0\]Solve for \(x\):- \((x+1)^2 = 0\) gives \(x = -1\)- \((x-2) = 0\) gives \(x = 2\)Thus, the critical points are \(-1\) and \(2\).

Step 2: Determine intervals

Divide the real number line into intervals using the critical points:1. \((-\infty, -1)\)2. \((-1, 2)\)3. \([2, \infty)\)

Step 3: Analyze the sign in each interval

For each interval, determine whether the expression \(-(x+1)^2(x-2)\) is positive or negative:- For \((-\infty, -1):\) Take \(x = -2\). Calculating: \(-(4)(-4) > 0\) (positive)- For \((-1, 2):\) Take \(x = 0\). Calculating: \(-(1)(-2) > 0\) (positive)- For \([2, \infty):\) Take \(x = 3\). Calculating: \(-(16)(1) < 0\) (negative)

Step 4: Consider the equality points

Check the behavior at the critical points:- At \(x = -1:\) \(-(0)(-3) = 0\) (included because of non-strict inequality)- At \(x = 2:\) \(-(16)(0) = 0\) (included because of non-strict inequality)

Step 5: Conclude the solution

Based on the sign analysis and checking signs at critical points, the solution set where the expression is non-negative is:\[x \in (-\infty, -1] \cup [-1, 2]\]

Key Concepts

Understanding Critical Points

When solving polynomial inequalities, finding critical points is a vital step. Critical points occur where the polynomial is equal to zero. This helps us figure out where the polynomial changes its sign. To find the critical points for the inequality \(-(x+1)^2(x-2) \geq 0 \), first set the polynomial equal to zero: \[ -(x+1)^2(x-2) = 0. \]Next, solve for \(x\) by setting each factor to zero:

• \((x+1)^2 = 0\) gives a critical point at \(x = -1\).
• \((x-2) = 0\) gives a critical point at \(x = 2\).

These are the points where the polynomial could change its sign. For this problem, the critical points are \(-1\) and \(2\). Understanding these points is crucial because they serve as boundaries when analyzing the behavior of the polynomial on different intervals.

Sign Analysis of Intervals

Once we have the critical points, we can perform sign analysis to determine the polynomial's behavior on different sections of the number line. The critical points divide the number line into distinct intervals. For the problem given, the intervals are:

• \((-fty, -1)\)
• \((-1, 2)\)
• \([2, fty)\)

Let's check the sign of the polynomial \(-(x+1)^2(x-2)\) in each interval. We select test points within each region to plug into the polynomial:- **Interval \((-fty, -1)\)**: Choose \(x = -2\). The expression simplifies to \(-(4)(-4) = 16\), which is positive.- **Interval \((-1, 2)\)**: Choose \(x = 0\). The expression simplifies to \(-(1)(-2) = 2\), which is also positive.- **Interval \([2, fty)\)**: Choose \(x = 3\). The expression simplifies to \(-(16)(1) = -16\), which is negative. By analyzing these results, we can determine where the polynomial is non-negative or non-positive.

Using Interval Notation

After conducting sign analysis, we need to express the solution using interval notation. Interval notation is a concise way of representing the set of solutions or the domain over which a polynomial inequality is satisfied. From the sign analysis, the non-negative intervals are found. These intervals need to include the critical points when the inequality is non-strict (\( \geq or \leq \)). For \( -(x+1)^2(x-2) \geq 0 \):- **Positive Interval**: \((-fty, -1)\) as the polynomial is positive.- **Positive Interval with critical point**: \([-1, 2]\) as the polynomial includes zero at both \(-1\) and \(2\).This leads to our solution being expressed as:\[x \, \in \, (-fty, -1] \, \cup \, [-1, 2] \]Interval notation provides a simple, readable method for conveying where inequalities are satisfied.

Finding Roots of Polynomials

In polynomial inequalities, identifying roots is essential. Roots are the values of \(x\) where the polynomial equals zero, which are instrumental for finding critical points. For our expression \(-(x+1)^2(x-2)\), the roots are calculated by solving when each factor equals zero:

• The first factor \((x + 1)^2 = 0\) gives the repeated root \(x = -1\).
• The second factor \((x - 2) = 0\) gives the root \(x = 2\).

These roots \(-1\) and \(2\) are pivotal as they determine where the polynomial potentially changes sign and helps us set the boundaries for our intervals. Recognizing and using these roots effectively allows for the creation of a structure for solving polynomial inequalities.