Solving Polynomial Inequalities A Step-by-Step Guide To (x+1)(x^2-3x+2)<0

Introduction to Polynomial Inequalities

When dealing with polynomial inequalities, the primary goal is to find the range of values for the variable x that satisfy the given inequality. Polynomial inequalities are mathematical expressions that involve polynomials and inequality symbols such as <, >, ≤, or ≥. Solving polynomial inequalities is a fundamental skill in algebra and calculus, with applications ranging from optimization problems to analyzing the behavior of functions. To effectively solve these inequalities, one must understand the behavior of polynomials, including their roots and how they change signs across different intervals. This article delves into the step-by-step process of solving the specific polynomial inequality $(x+1)(x^2-3x+2)<0$, providing a detailed explanation of the techniques and concepts involved. Mastering these techniques will equip you with the ability to tackle a wide variety of polynomial inequalities.

The process begins by factoring the polynomial expression, identifying the critical points (roots), and then testing intervals to determine where the inequality holds true. Each step is crucial to ensure an accurate solution. Polynomial inequalities often appear in various mathematical contexts, making their understanding essential for both academic and practical problem-solving scenarios. This guide will not only provide the solution to the given inequality but also offer insights into the general methodology for approaching such problems. By the end of this article, you will have a solid grasp of the methods and strategies needed to solve polynomial inequalities confidently.

Step 1: Factoring the Polynomial

The first crucial step in solving the polynomial inequality $(x+1)(x^2-3x+2)<0$ is to factor the polynomial expression completely. Factoring simplifies the inequality and allows us to identify the critical points, which are the values of x that make the polynomial equal to zero. These critical points divide the number line into intervals, which we will later test to determine the solution set. The given inequality already has one factor, $(x+1)$, but the quadratic expression $(x^2-3x+2)$ can be further factored.

To factor the quadratic $x^2-3x+2$, we look for two numbers that multiply to 2 and add to -3. These numbers are -1 and -2. Therefore, we can rewrite the quadratic expression as $(x-1)(x-2)$. Now, the inequality becomes:

$(x+1)(x-1)(x-2) < 0$

This factored form is essential because it allows us to easily identify the roots of the polynomial. The roots are the values of x that make the polynomial equal to zero, and they are the critical points that will help us determine the intervals where the inequality holds true. Factoring polynomial inequalities is a fundamental skill that simplifies the problem and sets the stage for the next steps in the solution process. By ensuring that the polynomial is fully factored, we can accurately identify the critical points and proceed with confidence.

Step 2: Identifying Critical Points

After factoring the polynomial inequality $(x+1)(x-1)(x-2) < 0$, the next crucial step is to identify the critical points. Critical points are the values of x that make the polynomial equal to zero. These points are significant because they divide the number line into intervals where the polynomial's sign (positive or negative) remains constant. In other words, the critical points are the boundaries where the polynomial can change its sign. To find the critical points, we set each factor of the polynomial equal to zero and solve for x.

From the factored form $(x+1)(x-1)(x-2)$, we have three factors:

1. $x+1 = 0$ implies $x = -1$
2. $x-1 = 0$ implies $x = 1$
3. $x-2 = 0$ implies $x = 2$

Thus, the critical points are $x = -1$, $x = 1$, and $x = 2$. These critical points divide the number line into four intervals: $(-\infty, -1)$, $(-1, 1)$, $(1, 2)$, and $(2, \infty)$. Identifying critical points is a fundamental aspect of solving polynomial inequalities, as it allows us to systematically analyze the behavior of the polynomial across different intervals. By pinpointing these points, we can determine the intervals where the inequality is satisfied by testing values within each interval. This step is vital for finding the solution set of the inequality.

Step 3: Testing Intervals

Once we have identified the critical points, the next step in solving the polynomial inequality $(x+1)(x-1)(x-2) < 0$ is to test each interval to determine where the inequality holds true. The critical points, which are $x = -1$, $x = 1$, and $x = 2$, divide the number line into four intervals: $(-\infty, -1)$, $(-1, 1)$, $(1, 2)$, and $(2, \infty)$. To test each interval, we select a test value within the interval and substitute it into the factored inequality.

1. Interval $(-\infty, -1)$: Choose $x = -2$. Substituting into the inequality gives $(-2+1)(-2-1)(-2-2) = (-1)(-3)(-4) = -12$, which is less than 0. So, the inequality holds true in this interval.
2. Interval $(-1, 1)$: Choose $x = 0$. Substituting into the inequality gives $(0+1)(0-1)(0-2) = (1)(-1)(-2) = 2$, which is not less than 0. So, the inequality does not hold true in this interval.
3. Interval $(1, 2)$: Choose $x = 1.5$. Substituting into the inequality gives $(1.5+1)(1.5-1)(1.5-2) = (2.5)(0.5)(-0.5) = -0.625$, which is less than 0. So, the inequality holds true in this interval.
4. Interval $(2, \infty)$: Choose $x = 3$. Substituting into the inequality gives $(3+1)(3-1)(3-2) = (4)(2)(1) = 8$, which is not less than 0. So, the inequality does not hold true in this interval.

By testing these intervals, we have determined the intervals where the polynomial is less than 0. This method of testing intervals is a fundamental technique in solving inequalities, as it allows us to systematically analyze the sign of the polynomial across different regions of the number line. Understanding this step is crucial for accurately determining the solution set of polynomial inequalities.

Step 4: Writing the Solution Set

After testing the intervals, we can now write the solution set for the polynomial inequality $(x+1)(x-1)(x-2) < 0$. We found that the inequality holds true in the intervals $(-\infty, -1)$ and $(1, 2)$. Therefore, the solution set consists of all x values within these intervals. In interval notation, the solution set is expressed as:

$(-\infty, -1) \cup (1, 2)$

This notation indicates that the solution includes all real numbers less than -1 and all real numbers between 1 and 2. The symbol “$\cup$” represents the union of the two intervals, meaning that all values in both intervals satisfy the inequality. Writing the solution set accurately is the final step in solving polynomial inequalities. It is essential to express the solution in a clear and concise manner, typically using interval notation or set-builder notation. The solution set represents all values of x that make the original inequality true.

In summary, solving polynomial inequalities involves factoring the polynomial, identifying critical points, testing intervals, and expressing the solution set. Each step is vital to ensure an accurate and complete solution. The solution set $(-\infty, -1) \cup (1, 2)$ is the final answer to the given polynomial inequality.

Conclusion: Mastering Polynomial Inequalities

In conclusion, mastering polynomial inequalities is a crucial skill in algebra and calculus. The process involves several key steps: factoring the polynomial, identifying critical points, testing intervals, and writing the solution set. By following these steps systematically, you can confidently solve a wide range of polynomial inequalities. The specific example of $(x+1)(x^2-3x+2)<0$ illustrates the method in detail, providing a clear and concise approach to solving such problems.

The initial step of factoring the polynomial simplifies the expression and makes it easier to identify critical points. These critical points are the roots of the polynomial and divide the number line into intervals. Identifying critical points is essential because they are the boundaries where the polynomial's sign can change. Testing these intervals allows us to determine where the polynomial is positive or negative, which is crucial for solving the inequality.

The process of testing intervals involves selecting a test value within each interval and substituting it into the factored inequality. This method helps us determine whether the inequality holds true in that interval. Finally, writing the solution set involves expressing the intervals where the inequality is satisfied using interval notation or set-builder notation.

The solution set for $(x+1)(x^2-3x+2)<0$ is $(-\infty, -1) \cup (1, 2)$, which represents all values of x that make the inequality true. By understanding and practicing these steps, you can effectively tackle any polynomial inequality. Polynomial inequalities have numerous applications in various fields, making their mastery a valuable asset for both academic and practical problem-solving scenarios. Therefore, a thorough understanding of these techniques will significantly enhance your mathematical proficiency.