Solving (x+1)(x^2-3x+2)<0 A Step-by-Step Guide
Polynomial inequalities, like the one presented – (x+1)(x^2-3x+2)<0 – are a fundamental topic in algebra, and mastering their solution is crucial for various applications in mathematics and related fields. This article provides a comprehensive guide to solving such inequalities, with a focus on the specific example of (x+1)(x^2-3x+2)<0. We'll break down the process into manageable steps, ensuring a clear understanding of the underlying concepts and techniques.
Understanding Polynomial Inequalities
Before diving into the solution, it's essential to grasp what a polynomial inequality represents. Unlike polynomial equations that seek specific values where the polynomial equals zero, polynomial inequalities aim to find intervals where the polynomial is either greater than, less than, greater than or equal to, or less than or equal to a certain value (often zero). These inequalities arise naturally in various contexts, such as optimization problems, where we might need to determine the range of inputs that yield a desired output range. The inequality (x+1)(x^2-3x+2)<0, for example, asks us to find all values of x for which the expression on the left-hand side results in a negative value. This requires us to analyze how the expression changes sign as x varies across the real number line. Understanding this fundamental concept is the first step in effectively tackling polynomial inequalities.
Step 1: Factor the Polynomial
The initial and crucial step in solving the inequality (x+1)(x^2-3x+2)<0 is to factor the polynomial completely. Factoring transforms the complex expression into a product of simpler factors, making it easier to analyze the sign changes. In our example, the quadratic expression x^2-3x+2 can be factored. To factor x^2-3x+2, we look for two numbers that multiply to 2 and add up to -3. These numbers are -1 and -2. Thus, we can rewrite the quadratic as (x-1)(x-2). Substituting this back into the original inequality, we get (x+1)(x-1)(x-2)<0. This factored form is significantly easier to work with, as it explicitly reveals the roots of the polynomial – the values of x that make the expression equal to zero. These roots play a pivotal role in determining the intervals where the polynomial is positive or negative. Factoring is not just a mechanical step; it's a crucial transformation that unlocks the structure of the polynomial and paves the way for a systematic solution.
Step 2: Identify the Critical Points
Once the polynomial is fully factored, the next crucial step in solving the inequality (x+1)(x-1)(x-2)<0 is to identify the critical points. These points are the roots of the polynomial, the values of x that make the expression equal to zero. In our factored form, (x+1)(x-1)(x-2), the critical points are readily apparent: x = -1, x = 1, and x = 2. These critical points are significant because they are the only values where the polynomial can change its sign. A polynomial can only transition from positive to negative or vice versa at its roots. This is a fundamental property of polynomials and forms the basis of our solution strategy. By identifying these critical points, we effectively divide the number line into intervals where the polynomial's sign remains constant. This division allows us to analyze the polynomial's behavior systematically, determining whether it's positive or negative within each interval. Therefore, identifying the critical points is not just a computational step; it's a conceptual leap that simplifies the problem into a series of manageable intervals.
Step 3: Create a Sign Chart
After identifying the critical points, constructing a sign chart is the next crucial step in solving the inequality (x+1)(x-1)(x-2)<0. A sign chart is a visual tool that helps us determine the sign of the polynomial expression in each interval defined by the critical points. It's a systematic way to organize our analysis and ensure we don't miss any possible solutions. To create the sign chart, we first draw a number line and mark the critical points (-1, 1, and 2) on it. These points divide the number line into four intervals: x < -1, -1 < x < 1, 1 < x < 2, and x > 2. Next, we consider each factor of the polynomial, (x+1), (x-1), and (x-2), and determine its sign in each interval. For example, (x+1) is negative when x < -1 and positive when x > -1. We repeat this process for the other factors, noting their signs in each interval. Finally, we multiply the signs of the factors in each interval to determine the sign of the entire polynomial expression. For instance, if an interval has two negative factors and one positive factor, the polynomial is positive in that interval. By systematically filling out the sign chart, we gain a clear picture of how the polynomial's sign changes across the number line. This visual representation makes it easy to identify the intervals where the inequality is satisfied.
Step 4: Test Values within Each Interval
An alternative approach, and a good way to verify the sign chart, is to test a value within each interval in the inequality (x+1)(x-1)(x-2)<0. This method involves selecting a test value from each of the intervals created by the critical points and substituting it into the factored polynomial expression. The sign of the result will indicate the sign of the polynomial throughout that interval. For example, in the interval x < -1, we could choose x = -2. Substituting this into the polynomial, we get (-2+1)(-2-1)(-2-2) = (-1)(-3)(-4) = -12, which is negative. This tells us that the polynomial is negative for all x < -1. We repeat this process for the other intervals: -1 < x < 1, 1 < x < 2, and x > 2, choosing a different test value for each. In the interval -1 < x < 1, we might choose x = 0, giving us (0+1)(0-1)(0-2) = (1)(-1)(-2) = 2, which is positive. For 1 < x < 2, we could choose x = 1.5, resulting in (1.5+1)(1.5-1)(1.5-2) = (2.5)(0.5)(-0.5) = -0.625, which is negative. Finally, for x > 2, we can choose x = 3, giving us (3+1)(3-1)(3-2) = (4)(2)(1) = 8, which is positive. By testing these values, we directly observe the sign of the polynomial in each interval, reinforcing our understanding and verifying the sign chart method. This approach is particularly useful for checking our work and ensuring the accuracy of our solution.
Step 5: Determine the Solution Set
With the sign chart or test values in hand, the final step in solving the inequality (x+1)(x-1)(x-2)<0 is to determine the solution set. The solution set consists of all the intervals where the polynomial satisfies the inequality – in this case, where the expression (x+1)(x-1)(x-2) is less than zero. Examining our sign chart or the results from our test values, we identify the intervals where the polynomial is negative. These are the intervals x < -1 and 1 < x < 2. It's crucial to pay attention to the inequality symbol. Since our inequality is strictly less than zero (<), we do not include the critical points themselves in the solution set, as these are the points where the polynomial equals zero. If the inequality were less than or equal to zero (≤), we would include the critical points. Therefore, the solution set for the inequality (x+1)(x-1)(x-2)<0 is the union of the intervals x < -1 and 1 < x < 2. We can express this solution set in interval notation as (-∞, -1) ∪ (1, 2). This notation clearly and concisely represents all the values of x that satisfy the given inequality. Determining the solution set is the culmination of our analysis, providing a complete and accurate answer to the problem.
Expressing the Solution
The solution to the polynomial inequality (x+1)(x^2-3x+2)<0, as we've determined through factoring, identifying critical points, using a sign chart, and testing values, is the set of all x values that make the expression negative. We've found that this occurs in the intervals x < -1 and 1 < x < 2. There are multiple ways to express this solution, each with its own advantages in terms of clarity and conciseness. One common way is to use interval notation, which we've already introduced. In interval notation, the solution is written as (-∞, -1) ∪ (1, 2). The parentheses indicate that the endpoints (-1 and 2) are not included in the solution, reflecting the strict inequality (<). The symbol ∪ represents the union of the two intervals, meaning that any value within either interval satisfies the inequality. Another way to express the solution is using inequality notation. In this form, we write the solution as x < -1 or 1 < x < 2. This notation directly states the conditions that x must satisfy. A third way to represent the solution is graphically. We can draw a number line and shade the intervals (-∞, -1) and (1, 2), visually indicating the solution set. An open circle at -1 and 2 would further emphasize that these points are not included. Choosing the most appropriate way to express the solution depends on the context and the desired level of detail. However, regardless of the notation used, it's crucial to ensure that the solution accurately represents all values of x that satisfy the original inequality.
Conclusion
Solving polynomial inequalities like (x+1)(x^2-3x+2)<0 involves a systematic approach that combines algebraic manipulation with careful analysis. By factoring the polynomial, identifying critical points, creating a sign chart (or testing values), and determining the solution set, we can effectively solve a wide range of polynomial inequalities. The key is to break down the problem into manageable steps and understand the underlying concepts. Mastering these techniques not only equips you to solve specific problems but also provides a solid foundation for more advanced mathematical concepts. Polynomial inequalities are a cornerstone of algebra and calculus, and the ability to solve them confidently is an invaluable skill.
