Solving Polynomial Inequality $-3(x+1)(x-2)(x+6) \geq 0$ A Step-by-Step Guide

Polynomial inequalities are a fundamental topic in algebra and calculus, crucial for understanding the behavior of functions and solving a variety of mathematical problems. This article delves into a step-by-step guide on how to solve the polynomial inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$. We will explore the key concepts, methodologies, and practical techniques required to tackle such problems effectively. Whether you are a student learning algebra or a professional looking to refresh your skills, this comprehensive guide will provide you with the necessary tools to solve polynomial inequalities with confidence.

Polynomial inequalities, like the one we are addressing, involve finding the range of values for a variable that satisfy a given inequality. These inequalities are particularly important in various fields, including engineering, economics, and computer science, where understanding the boundaries and conditions of solutions is critical. By mastering the methods to solve these inequalities, you enhance your problem-solving capabilities and gain a deeper insight into mathematical functions.

The inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$ is a quintessential example that encompasses multiple factors and a negative coefficient, making it an excellent case study. We will break down the solution process into manageable steps, starting from identifying the roots of the polynomial to determining the intervals where the inequality holds true. This article aims to not only provide the solution but also to explain the underlying principles, ensuring a thorough understanding of the topic.

At its core, polynomial inequalities involve comparing a polynomial expression to zero. These inequalities can take various forms, such as greater than, less than, greater than or equal to, or less than or equal to. Solving them requires identifying the values of the variable that make the inequality true. The polynomial inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$ falls into this category, and to solve it, we must find the values of $x$ that satisfy the condition where the expression is greater than or equal to zero.

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the expression. For instance, in our inequality, the polynomial is of degree 3 because when expanded, the highest power of $x$ will be $x^3$. Understanding the degree of a polynomial is crucial as it influences the number of possible roots and the overall shape of the polynomial function's graph.

The roots of a polynomial, also known as zeros, are the values of the variable that make the polynomial equal to zero. In the context of inequalities, the roots play a pivotal role because they are the points at which the polynomial can change its sign (from positive to negative or vice versa). Identifying these roots is the first critical step in solving a polynomial inequality. For the given inequality, the roots can be found by setting each factor to zero: $x+1=0$, $x-2=0$, and $x+6=0$. These roots divide the number line into intervals, which we will analyze to determine where the inequality holds.

Another key concept in solving polynomial inequalities is the sign analysis of the polynomial across different intervals. Since the polynomial can only change signs at its roots, we can test a value within each interval to determine the sign of the polynomial in that interval. This method allows us to systematically identify the intervals where the polynomial is either positive or negative, which is essential for solving the inequality. The sign analysis, combined with the knowledge of the roots, provides a clear pathway to the solution.

In summary, solving polynomial inequalities involves a combination of algebraic techniques and logical reasoning. Understanding the nature of polynomials, identifying their roots, and performing sign analysis are the fundamental steps. These concepts are not only applicable to this specific inequality but also form a strong foundation for tackling more complex problems in algebra and calculus. In the following sections, we will apply these principles to solve the inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$ in detail.

To effectively solve the polynomial inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$, we will follow a structured approach that includes identifying the roots, creating intervals, testing points within those intervals, and expressing the solution in interval notation. This step-by-step method ensures clarity and accuracy in the solution process.

Step 1: Identify the Roots

The first step in solving a polynomial inequality is to find the roots of the polynomial. The roots are the values of $x$ that make the polynomial equal to zero. For the inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$, we set each factor equal to zero and solve for $x$:

• $x + 1 = 0$ gives $x = -1$
• $x - 2 = 0$ gives $x = 2$
• $x + 6 = 0$ gives $x = -6$

Thus, the roots of the polynomial are $x = -6$, $x = -1$, and $x = 2$. These roots are crucial because they divide the number line into intervals where the polynomial's sign may change.

Step 2: Create Intervals

Once the roots are identified, the next step is to divide the number line into intervals using these roots. The roots $-6$, $-1$, and $2$ create four intervals:

• $(-${[infinity]], -6)$
• $(-6, -1)$
• $(-1, 2)$
• $(2, \+\[\[infinity]])$

These intervals represent the regions on the number line where the polynomial's value will either be consistently positive or consistently negative. The roots themselves are the boundary points where the polynomial can change its sign.

Step 3: Test Points within Intervals

To determine the sign of the polynomial in each interval, we select a test point from each interval and substitute it into the polynomial expression $-3(x+1)(x-2)(x+6)$. This will tell us whether the polynomial is positive or negative in that interval.

1. Interval $(-\[\[infinity]], -6)$: Choose $x = -7$ $-3((-7)+1)((-7)-2)((-7)+6) = -3(-6)(-9)(-1) = -162$. The polynomial is negative.
2. Interval $(-6, -1)$: Choose $x = -2$ $-3((-2)+1)((-2)-2)((-2)+6) = -3(-1)(-4)(4) = -48$. The polynomial is negative.
3. Interval $(-1, 2)$: Choose $x = 0$ $-3((0)+1)((0)-2)((0)+6) = -3(1)(-2)(6) = 36$. The polynomial is positive.
4. Interval $(2, \+\[\[infinity]])$: Choose $x = 3$ $-3((3)+1)((3)-2)((3)+6) = -3(4)(1)(9) = -108$. The polynomial is negative.

Step 4: Determine the Solution

We are solving the inequality -3(x+1)(x-2)(x+6) \[\geq 0}$, which means we are looking for intervals where the polynomial is either positive or equal to zero. Based on our test points, the polynomial is positive in the interval $(-1, 2)$. Additionally, the polynomial is equal to zero at the roots $x = -6$, $x = -1$, and $x = 2$. Therefore, we include these roots in our solution.

Step 5: Write the Solution in Interval Notation

Finally, we express the solution in interval notation. Since we include the roots where the polynomial is equal to zero, we use square brackets. The intervals where the polynomial is greater than or equal to zero are:

• $[-6, -1]$
• $[2, 2]$

Combining these, the solution in interval notation is $[-6,-1] \cup [2,2]$.

By following these steps, we have systematically solved the polynomial inequality $-3(x+1)(x-2)(x+6) ${\geq 0}$$. The solution includes the intervals where the polynomial is positive or zero, providing a comprehensive understanding of the inequality's behavior.

Once we have determined the intervals where the polynomial inequality holds true, the final step is to express the solution using interval notation. Interval notation is a concise way to represent sets of real numbers and is particularly useful when dealing with inequalities. This notation uses brackets and parentheses to indicate whether the endpoints of an interval are included or excluded from the solution set.

To effectively use interval notation, it is crucial to understand the meaning of the symbols involved. Square brackets, [ and ], indicate that the endpoint is included in the interval. This is used when the inequality includes an