Solving $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$ A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the polynomial inequality $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$. Polynomial inequalities, a fundamental topic in algebra, involve determining the range of values for a variable that satisfy an inequality involving polynomial expressions. This particular inequality is a cubic inequality, which means it involves a polynomial of degree three. Solving such inequalities often requires a combination of algebraic manipulation, factoring techniques, and sign analysis. The solution will be expressed in interval notation, a standard way to represent sets of real numbers. Understanding how to solve polynomial inequalities is crucial for various applications in mathematics, including calculus, optimization problems, and real-world modeling. We'll break down each step, providing clear explanations and strategies to tackle this problem effectively.

Polynomial inequalities play a crucial role in various mathematical contexts, from determining the domain of functions to solving optimization problems. Mastering the techniques required to solve these inequalities provides a strong foundation for more advanced mathematical concepts. In this comprehensive guide, we will walk through the detailed steps of solving the given cubic inequality, ensuring that you understand each phase of the process. Our approach will include simplifying the inequality, factoring the polynomial, identifying critical points, and analyzing the intervals to determine the solution set. By the end of this article, you will be equipped with the knowledge and skills necessary to confidently tackle similar polynomial inequalities. Let's embark on this mathematical journey and unravel the solution to this engaging problem.

Before we dive into the specifics, it's important to grasp the underlying principles of inequality solving. Unlike equations, where we seek specific values that make the expressions equal, inequalities ask for a range of values that satisfy a certain condition, such as being greater than, less than, greater than or equal to, or less than or equal to. When dealing with polynomials, this often involves finding the intervals where the polynomial's value is positive or negative. These intervals are typically determined by the roots (or zeros) of the polynomial, which act as critical points that divide the number line into segments. Each segment must then be tested to see if it satisfies the inequality. With these principles in mind, we can confidently proceed to solve our inequality.

Step 1: Simplify the Inequality

The first step in solving any inequality, especially one involving polynomials, is to simplify the expression. This involves removing parentheses, combining like terms, and rearranging the inequality to a standard form. For our given inequality, $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$, we start by distributing the negative sign across the terms inside the parentheses. This yields $4x^3 - 12x - 8 + 12x^2 + 8x - 4 > 0$. The next step is to combine like terms. We have terms involving $x^3$, $x^2$, $x$, and constants. Combining the terms, we get $4x^3 + 12x^2 - 12x + 8x - 8 - 4 > 0$. Simplifying further, we have $4x^3 + 12x^2 - 4x - 12 > 0$. This simplified form is much easier to work with and allows us to proceed to the next steps.

Simplifying an inequality not only makes it more manageable but also reveals its underlying structure. By removing parentheses and combining like terms, we can see the polynomial in its simplest form, which is crucial for the subsequent steps of factoring and finding roots. The simplified form $4x^3 + 12x^2 - 4x - 12 > 0$ allows us to identify the coefficients and the degree of the polynomial, which are essential for choosing the appropriate factoring techniques. Furthermore, simplification reduces the likelihood of making errors in the later steps of the solution process. With the inequality in its simplified form, we are well-prepared to move on to the next challenge: factoring the polynomial. This will allow us to identify the critical points that define the intervals where the inequality holds.

By starting with simplification, we adhere to a systematic approach that is vital in solving mathematical problems, particularly those involving inequalities. The simplified polynomial inequality is now in a standard form, making it easier to apply factoring techniques. Factoring is a critical step because it helps us find the roots of the polynomial, which are the points where the polynomial equals zero. These roots serve as the boundaries for the intervals we will analyze to determine where the inequality holds true. Essentially, simplification sets the stage for factoring, and factoring sets the stage for finding solutions. This methodical progression is key to successfully solving complex mathematical problems. With the simplified inequality in hand, we are ready to tackle the task of factoring, which is the next crucial step in our solution journey.

Step 2: Factor the Polynomial

Now that we have the simplified inequality $4x^3 + 12x^2 - 4x - 12 > 0$, the next step is to factor the polynomial. Factoring is a crucial technique for solving polynomial inequalities because it allows us to express the polynomial as a product of simpler factors. These factors help us identify the roots of the polynomial, which are the points where the polynomial equals zero. To factor this cubic polynomial, we can use the method of factoring by grouping. First, we group the terms: $(4x^3 + 12x^2) + (-4x - 12)$. Next, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out $4x^2$, giving us $4x^2(x + 3)$. From the second group, we can factor out $-4$, giving us $-4(x + 3)$. Now we have $4x^2(x + 3) - 4(x + 3)$. Notice that $(x + 3)$ is a common factor in both terms, so we can factor it out, resulting in $(x + 3)(4x^2 - 4)$. We can further simplify the second factor by factoring out 4, giving us $4(x^2 - 1)$. The term $(x^2 - 1)$ is a difference of squares, which can be factored as $(x - 1)(x + 1)$. Therefore, the fully factored form of the polynomial is $4(x + 3)(x - 1)(x + 1)$.

Factoring the polynomial is an essential step because it transforms a complex expression into a product of simpler expressions, making it easier to find the roots. These roots are critical points that divide the number line into intervals, which we will analyze to determine where the inequality is satisfied. Factoring by grouping, as demonstrated in this case, is a powerful technique for polynomials with four terms. The process involves identifying common factors within groups of terms and then factoring out those common factors. This often reveals a common binomial factor that can be factored out, leading to a fully factored polynomial. The factored form, $4(x + 3)(x - 1)(x + 1)$, now allows us to easily identify the roots of the polynomial, which are the values of $x$ that make the polynomial equal to zero. These roots are the key to solving the inequality.

In summary, the factoring process has transformed our cubic polynomial into a product of linear factors. Each linear factor corresponds to a root of the polynomial. The factored form, $4(x + 3)(x - 1)(x + 1)$, is much easier to analyze than the original polynomial. By setting each factor equal to zero, we can find the roots of the polynomial, which are the critical points for our inequality. These critical points are $x = -3$, $x = -1$, and $x = 1$. These roots partition the number line into intervals, and we will analyze the sign of the polynomial within each interval to determine where the inequality $4(x + 3)(x - 1)(x + 1) > 0$ holds true. The ability to factor polynomials efficiently is a fundamental skill in algebra, and this step has brought us closer to solving the inequality.

Step 3: Find the Critical Points

With the polynomial factored as $4(x + 3)(x - 1)(x + 1)$, the next crucial step is to find the critical points. Critical points are the values of $x$ that make the polynomial equal to zero. These points are essential because they divide the number line into intervals, and within each interval, the polynomial will either be entirely positive or entirely negative. To find the critical points, we set each factor equal to zero and solve for $x$. Setting $x + 3 = 0$, we find $x = -3$. Setting $x - 1 = 0$, we find $x = 1$. And setting $x + 1 = 0$, we find $x = -1$. Thus, the critical points are $x = -3$, $x = -1$, and $x = 1$. These critical points are the roots of the polynomial and play a vital role in determining the solution set of the inequality.

Finding the critical points is a pivotal step in solving polynomial inequalities. These points are the values of $x$ where the polynomial changes sign, effectively acting as boundaries between intervals where the polynomial is positive and intervals where it is negative. By setting each factor of the factored polynomial to zero, we identify these critical points. In our case, the critical points $x = -3$, $x = -1$, and $x = 1$ divide the number line into four intervals: $(-\infty, -3)$, $(-3, -1)$, $(-1, 1)$, and $(1, \infty)$. The sign of the polynomial remains constant within each interval, so we only need to test a single point within each interval to determine the sign of the polynomial throughout that interval. This simplifies the process of solving the inequality significantly.

Once the critical points are identified, they provide a clear roadmap for the next phase of the solution. The number line is now partitioned into intervals, and we know that the polynomial's sign will not change within each interval. This knowledge allows us to create a sign chart, which is a visual tool that helps us organize our analysis. The critical points are marked on the number line, and we choose a test value from each interval to evaluate the polynomial. The sign of the polynomial at each test value will determine the sign of the polynomial throughout the entire interval. In the next step, we will use these critical points to construct a sign chart and analyze the intervals to find the solution set for our inequality. The critical points are the foundation upon which we build the solution.

Step 4: Create a Sign Chart and Test Intervals

Having identified the critical points $x = -3$, $x = -1$, and $x = 1$, the next step is to create a sign chart. A sign chart is a visual tool that helps us determine the sign of the polynomial in each interval created by the critical points. We draw a number line and mark the critical points on it. These points divide the number line into four intervals: $(-\infty, -3)$, $(-3, -1)$, $(-1, 1)$, and $(1, \infty)$. We then choose a test value within each interval and evaluate the factored polynomial $4(x + 3)(x - 1)(x + 1)$ at that value. The sign of the polynomial at the test value will be the sign of the polynomial throughout the entire interval.

For the interval $(-\infty, -3)$, we can choose a test value of $x = -4$. Evaluating the polynomial at $x = -4$, we get $4(-4 + 3)(-4 - 1)(-4 + 1) = 4(-1)(-5)(-3) = -60$, which is negative. Therefore, the polynomial is negative in the interval $(-\infty, -3)$. For the interval $(-3, -1)$, we can choose a test value of $x = -2$. Evaluating the polynomial at $x = -2$, we get $4(-2 + 3)(-2 - 1)(-2 + 1) = 4(1)(-3)(-1) = 12$, which is positive. Therefore, the polynomial is positive in the interval $(-3, -1)$. For the interval $(-1, 1)$, we can choose a test value of $x = 0$. Evaluating the polynomial at $x = 0$, we get $4(0 + 3)(0 - 1)(0 + 1) = 4(3)(-1)(1) = -12$, which is negative. Therefore, the polynomial is negative in the interval $(-1, 1)$. Finally, for the interval $(1, \infty)$, we can choose a test value of $x = 2$. Evaluating the polynomial at $x = 2$, we get $4(2 + 3)(2 - 1)(2 + 1) = 4(5)(1)(3) = 60$, which is positive. Therefore, the polynomial is positive in the interval $(1, \infty)$.

The sign chart provides a clear picture of where the polynomial is positive and where it is negative. The test values are strategically chosen to represent each interval, ensuring that the sign of the polynomial at the test value accurately reflects the sign of the polynomial throughout the interval. The calculations we performed reveal that the polynomial is negative in the intervals $(-\infty, -3)$ and $(-1, 1)$, and positive in the intervals $(-3, -1)$ and $(1, \infty)$. Since we are looking for the intervals where the polynomial is greater than zero, we are interested in the intervals where the polynomial is positive. The sign chart is an invaluable tool for visualizing and organizing this information, making it easier to identify the solution set. With the sign chart completed, we are now ready to express the solution in interval notation.

Step 5: Express the Solution in Interval Notation

After creating the sign chart and testing the intervals, we have determined that the polynomial $4(x + 3)(x - 1)(x + 1)$ is greater than zero in the intervals $(-3, -1)$ and $(1, \infty)$. The original inequality is $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$, which we simplified and factored to $4(x + 3)(x - 1)(x + 1) > 0$. The solution to this inequality is the set of all $x$ values that make the polynomial positive. Based on our sign chart analysis, these values lie in the intervals $(-3, -1)$ and $(1, \infty)$. Therefore, the solution in interval notation is $(-3, -1) \cup (1, \infty)$. This notation represents the union of the two intervals, indicating that the solution includes all real numbers between $-3$ and $-1$ (excluding $-3$ and $-1$) and all real numbers greater than $1$.

Expressing the solution in interval notation is the final step in solving the inequality. Interval notation is a concise and standard way of representing sets of real numbers. The solution $(-3, -1) \cup (1, \infty)$ accurately captures the range of $x$ values that satisfy the inequality. The parentheses indicate that the endpoints $-3$, $-1$, and $1$ are not included in the solution, which is consistent with the strict inequality ($>$) in the original problem. If the inequality were greater than or equal to ($\geq$), we would use square brackets to include the endpoints in the solution. The union symbol ($\cup$) indicates that the solution set is the combination of the two intervals. This notation provides a clear and unambiguous way to communicate the solution.

In conclusion, the solution to the inequality $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$ in interval notation is $(-3, -1) \cup (1, \infty)$. We arrived at this solution by systematically simplifying the inequality, factoring the polynomial, identifying the critical points, creating a sign chart, and testing intervals. Each step was crucial in unraveling the problem and arriving at the correct answer. This process demonstrates the power of algebraic techniques in solving complex inequalities and highlights the importance of a structured approach. Understanding and applying these techniques will enable you to solve a wide range of polynomial inequalities with confidence.

Summary

In this comprehensive guide, we have successfully solved the cubic inequality $4x^3 - 12x - 8 - (-12x^2 - 8x + 4) > 0$. Our systematic approach involved several key steps, each building upon the previous one to lead us to the final solution. First, we simplified the inequality by removing parentheses and combining like terms, resulting in $4x^3 + 12x^2 - 4x - 12 > 0$. Next, we factored the polynomial using the method of factoring by grouping, obtaining the factored form $4(x + 3)(x - 1)(x + 1) > 0$. We then identified the critical points by setting each factor equal to zero, which gave us $x = -3$, $x = -1$, and $x = 1$. These critical points divided the number line into four intervals. To determine the sign of the polynomial in each interval, we created a sign chart and tested a value within each interval. Finally, based on the sign chart analysis, we expressed the solution in interval notation as $(-3, -1) \cup (1, \infty)$.

Each step in the solution process is critical and reinforces fundamental algebraic principles. Simplifying the inequality makes it more manageable and easier to work with. Factoring the polynomial allows us to identify the roots, which are the critical points that divide the number line into intervals. Identifying critical points is crucial because these points are where the polynomial changes sign. Creating a sign chart is a visual and organized way to analyze the sign of the polynomial in each interval. Testing intervals ensures that we accurately determine where the polynomial satisfies the inequality. And finally, expressing the solution in interval notation provides a concise and standard way to represent the set of all $x$ values that satisfy the inequality.

By following this structured approach, we can effectively solve polynomial inequalities of various degrees. The ability to solve these types of inequalities is essential in many areas of mathematics, including calculus, optimization problems, and real-world modeling. The techniques and strategies discussed in this guide provide a solid foundation for tackling more complex mathematical challenges. The solution $(-3, -1) \cup (1, \infty)$ represents the range of $x$ values that make the original inequality true, and our step-by-step solution demonstrates how to systematically arrive at this answer. This guide serves as a valuable resource for anyone seeking to improve their understanding and skills in solving polynomial inequalities.