Solving Polynomial Inequality X³ - 4x² + 3x ≤ 0 A Comprehensive Guide

In the realm of mathematics, solving inequalities is a fundamental skill, particularly when dealing with polynomials. This article delves into the intricacies of solving the polynomial inequality x³ - 4x² + 3x ≤ 0. We will explore the necessary steps, underlying concepts, and techniques to tackle such problems effectively. Whether you're a student grappling with algebra or a math enthusiast seeking to enhance your problem-solving abilities, this guide will provide a comprehensive understanding of polynomial inequalities.

Before diving into the specifics of x³ - 4x² + 3x ≤ 0, it's crucial to grasp the basics of polynomial inequalities. A polynomial inequality is a mathematical statement comparing a polynomial expression to a value (in this case, 0) using inequality symbols such as <, >, ≤, or ≥. The solutions to these inequalities are the values of the variable (here, x) that make the inequality true. Solving polynomial inequalities involves finding the intervals on the number line where the polynomial's value satisfies the given condition.

The general approach involves several key steps:

1. Rearrange the inequality: Ensure that one side of the inequality is 0. This sets the stage for analyzing the polynomial's behavior around its roots.
2. Factor the polynomial: Factoring the polynomial helps identify its roots, which are the points where the polynomial's value equals 0. These roots are critical in determining the intervals where the polynomial changes its sign.
3. Find the roots: Determine the values of x that make the polynomial equal to 0. These roots are the boundary points that divide the number line into intervals.
4. Create a sign chart: Construct a number line and mark the roots. Choose test values within each interval and evaluate the polynomial. This will reveal whether the polynomial is positive or negative in each interval.
5. Determine the solution: Based on the sign chart and the inequality symbol, identify the intervals that satisfy the inequality. Remember to include or exclude the roots depending on whether the inequality is strict (<, >) or non-strict (≤, ≥).

Now, let's apply these steps to solve the inequality x³ - 4x² + 3x ≤ 0. This example will demonstrate the practical application of the general principles outlined above.

Step 1: Rearrange the Inequality

In this case, the inequality is already in the desired form, with 0 on one side:

x³ - 4x² + 3x ≤ 0

This simplifies our task, allowing us to proceed directly to factoring the polynomial.

Step 2: Factor the Polynomial

To factor the polynomial x³ - 4x² + 3x, we first look for a common factor. Notice that x is a common factor in all terms. Factoring out x gives us:

x(x² - 4x + 3) ≤ 0

Next, we factor the quadratic expression x² - 4x + 3. We are looking for two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3. Therefore, we can factor the quadratic as:

x(x - 1)(x - 3) ≤ 0

This completes the factorization, giving us the polynomial in a form that is easy to analyze.

Step 3: Find the Roots

To find the roots, we set each factor equal to 0 and solve for x:

• x = 0
• x - 1 = 0 => x = 1
• x - 3 = 0 => x = 3

So, the roots are x = 0, x = 1, and x = 3. These roots divide the number line into four intervals, which we will analyze in the next step.

Step 4: Create a Sign Chart

Now, we construct a sign chart to determine the sign of the polynomial in each interval. We mark the roots 0, 1, and 3 on the number line. This divides the number line into the following intervals:

• (-∞, 0)
• (0, 1)
• (1, 3)
• (3, ∞)

We choose a test value in each interval and evaluate the polynomial x(x - 1)(x - 3):

• Interval (-∞, 0): Let's choose x = -1. Then, (-1)(-1 - 1)(-1 - 3) = (-1)(-2)(-4) = -8, which is negative.
• Interval (0, 1): Let's choose x = 0.5. Then, (0.5)(0.5 - 1)(0.5 - 3) = (0.5)(-0.5)(-2.5) = 0.625, which is positive.
• Interval (1, 3): Let's choose x = 2. Then, (2)(2 - 1)(2 - 3) = (2)(1)(-1) = -2, which is negative.
• Interval (3, ∞): Let's choose x = 4. Then, (4)(4 - 1)(4 - 3) = (4)(3)(1) = 12, which is positive.

We can summarize these results in a sign chart:

Step 5: Determine the Solution

We are looking for the intervals where x(x - 1)(x - 3) ≤ 0. This means we want the intervals where the polynomial is negative or equal to 0. From the sign chart, we see that the polynomial is negative in the intervals (-∞, 0) and (1, 3). It is equal to 0 at the roots x = 0, x = 1, and x = 3.

Therefore, the solution to the inequality is the union of the intervals (-∞, 0] and [1, 3]. We include the roots because the inequality is non-strict (≤).

In conclusion, the solution to the polynomial inequality x³ - 4x² + 3x ≤ 0 is x ∈ (-∞, 0] ∪ [1, 3]. This comprehensive solution was achieved by systematically factoring the polynomial, finding its roots, constructing a sign chart, and interpreting the results. Mastering these steps empowers you to tackle a wide range of polynomial inequalities with confidence. The ability to solve such inequalities is a crucial skill in various mathematical contexts, making this knowledge invaluable for students and enthusiasts alike.

By understanding the underlying principles and practicing these techniques, you can confidently approach and solve polynomial inequalities, enhancing your mathematical proficiency and problem-solving skills. Remember to always double-check your work and ensure that your solution aligns with the given inequality. With dedication and practice, you can master the art of solving polynomial inequalities and unlock new levels of mathematical understanding.