Solving Polynomial Inequality $x^3 + 3x^2 \geq -x - 3$ A Step-by-Step Guide

Introduction

In this article, we will walk through the process of solving the polynomial inequality $x^3 + 3x^2 \geq -x - 3$. Inequalities, especially those involving polynomials, are a fundamental topic in algebra and calculus. Understanding how to solve them is crucial for various applications in mathematics, science, and engineering. We will employ factoring techniques and interval analysis to find the solution set. Our goal is to provide a clear, step-by-step explanation that enables readers to confidently tackle similar problems. The solution will be expressed in interval notation, a standard way to represent sets of real numbers. Let’s dive into the detailed steps to find the solution.

Understanding Polynomial Inequalities

Before we delve into solving the specific inequality, let's briefly discuss polynomial inequalities in general. Polynomial inequalities involve comparing a polynomial expression to another value, typically zero. The solutions to these inequalities are the intervals of $x$ values that make the inequality true. Unlike equations, which have discrete solutions, inequalities often have solution sets that are ranges of values. Solving polynomial inequalities usually involves rearranging the inequality, factoring the polynomial, finding critical points, and testing intervals. Critical points are the values of $x$ where the polynomial equals zero or is undefined. These points divide the number line into intervals, and within each interval, the polynomial's sign remains constant. By testing a value from each interval, we can determine whether that interval is part of the solution set. This systematic approach allows us to find all $x$ values that satisfy the given inequality. The ability to solve polynomial inequalities is a key skill in mathematical analysis and is applied in various contexts, including optimization problems and the study of function behavior.

Step-by-Step Solution

1. Rearrange the Inequality

Our first step in solving the inequality $x^3 + 3x^2 \geq -x - 3$ is to rearrange it so that one side is zero. This is a standard technique that simplifies the factoring process. We achieve this by adding $x$ and $3$ to both sides of the inequality:

$x^3 + 3x^2 + x + 3 \geq 0$

This rearrangement sets the stage for factoring, which is crucial for identifying the critical points of the inequality. By having zero on one side, we can easily analyze the sign of the polynomial expression. This step is a fundamental part of solving inequalities, as it transforms the problem into a form that is easier to handle. Now that we have the inequality in the desired form, we can proceed to the next step, which involves factoring the polynomial expression.

2. Factor the Polynomial

Now, we need to factor the polynomial $x^3 + 3x^2 + x + 3$. Factoring is the process of breaking down a polynomial into a product of simpler polynomials. In this case, we can use factoring by grouping. Group the first two terms and the last two terms:

$(x^3 + 3x^2) + (x + 3) \geq 0$

Next, factor out the greatest common factor (GCF) from each group. From the first group, $x^2$ is the GCF, and from the second group, it is $1$:

$x^2(x + 3) + 1(x + 3) \geq 0$

Now, we can see that $(x + 3)$ is a common factor in both terms. Factor it out:

$(x + 3)(x^2 + 1) \geq 0$

The polynomial is now factored. This factorization is a critical step because it allows us to identify the roots of the polynomial, which are essential for finding the intervals where the inequality holds true. The expression $(x^2 + 1)$ is always positive for real values of $x$, since $x^2$ is non-negative, and adding $1$ makes it strictly positive. Thus, it does not contribute to the sign changes of the inequality. The factor $(x + 3)$ is the key to determining the solution. Factoring the polynomial transforms a complex expression into a simpler form that reveals its structure and behavior.

3. Find Critical Points

Critical points are the values of $x$ that make the expression equal to zero. These points divide the number line into intervals, where the polynomial's sign remains constant within each interval. To find the critical points for our factored inequality $(x + 3)(x^2 + 1) \geq 0$, we set each factor equal to zero and solve for $x$:

$x + 3 = 0$ gives $x = -3$

The factor $(x^2 + 1)$ has no real roots because $x^2$ is always non-negative, and adding $1$ makes it strictly positive. Therefore, it never equals zero for any real value of $x$. Thus, the only critical point is $x = -3$. This single critical point divides the number line into two intervals: $(-\infty, -3)$ and $(-3, \infty)$. These intervals are crucial for the next step, where we will test values within each interval to determine where the inequality holds true. Identifying critical points is a cornerstone of solving inequalities, as it provides the boundaries for the intervals we need to analyze.

4. Test Intervals

Now that we have identified the critical point $x = -3$, we need to test the intervals $(-\infty, -3)$ and $(-3, \infty)$ to determine where the inequality $(x + 3)(x^2 + 1) \geq 0$ holds. We will choose a test value within each interval and substitute it into the factored inequality. The sign of the result will tell us whether the interval is part of the solution.

Interval $(-\infty, -3)$

Choose a test value, for example, $x = -4$. Substitute it into the inequality:

$(-4 + 3)((-4)^2 + 1) = (-1)(16 + 1) = (-1)(17) = -17$

Since $-17 < 0$, the inequality is not satisfied in this interval.

Interval $(-3, \infty)$

Choose a test value, for example, $x = 0$. Substitute it into the inequality:

$(0 + 3)(0^2 + 1) = (3)(1) = 3$

Since $3 > 0$, the inequality is satisfied in this interval.

Additionally, we need to consider the critical point $x = -3$. When $x = -3$, the inequality becomes:

$(-3 + 3)((-3)^2 + 1) = (0)(10) = 0$

Since $0 \geq 0$, the critical point $x = -3$ is also part of the solution.

Testing intervals is a crucial step in solving inequalities because it allows us to determine the ranges of values that satisfy the given condition. By choosing representative values from each interval, we can efficiently analyze the behavior of the inequality. The results of these tests will form the basis for writing the solution in interval notation.

5. Write the Solution in Interval Notation

Based on the interval testing, we found that the inequality $(x + 3)(x^2 + 1) \geq 0$ is satisfied in the interval $(-3, \infty)$ and at the critical point $x = -3$. Therefore, the solution includes all values greater than or equal to $-3$. In interval notation, this is represented as $[-3, \infty)$. The square bracket indicates that $-3$ is included in the solution set, while the parenthesis indicates that infinity is not included, as it is a concept rather than a number.

Conclusion

In summary, to solve the inequality $x^3 + 3x^2 \geq -x - 3$, we followed these steps:

1. Rearranged the inequality to have zero on one side: $x^3 + 3x^2 + x + 3 \geq 0$.
2. Factored the polynomial: $(x + 3)(x^2 + 1) \geq 0$.
3. Found the critical points: $x = -3$.
4. Tested the intervals $(-\infty, -3)$ and $(-3, \infty)$ to determine where the inequality holds.
5. Wrote the solution in interval notation: $[-3, \infty)$.

This detailed process illustrates how to solve polynomial inequalities systematically. By rearranging, factoring, identifying critical points, testing intervals, and expressing the solution in interval notation, we can confidently solve a wide range of inequalities. Understanding these techniques is crucial for success in algebra, calculus, and various fields that rely on mathematical analysis. The solution $[-3, \infty)$ represents all real numbers greater than or equal to $-3$, which satisfy the original inequality.