Solving Polynomial Inequality X^3 - 4x^2 + 3x ≤ 0 A Step-by-Step Guide

In this article, we will delve into the process of solving the polynomial inequality $x^3 - 4x^2 + 3x gtr 0$. This involves finding the range of values for $x$ that satisfy the given inequality. Polynomial inequalities are a fundamental topic in algebra and calculus, and understanding how to solve them is crucial for various mathematical applications. We will break down the problem step by step, employing techniques such as factoring, finding critical points, and testing intervals to determine the solution set. By the end of this discussion, you will have a comprehensive understanding of how to tackle similar polynomial inequalities.

To begin solving the inequality $x^3 - 4x^2 + 3x gtr 0$, the first crucial step is to factor the polynomial. Factoring simplifies the expression and allows us to identify the critical points, which are the values of $x$ where the polynomial equals zero. These critical points are essential for determining the intervals where the polynomial is either positive or negative.

Starting with the given polynomial, $x^3 - 4x^2 + 3x$, we can observe that $x$ is a common factor in all the terms. Factoring out $x$, we get:

$x(x^2 - 4x + 3) gtr 0$

Now, we need to factor the quadratic expression $x^2 - 4x + 3$. 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:

$x(x - 1)(x - 3) gtr 0$

Thus, the factored form of the polynomial is $x(x - 1)(x - 3)$. This factorization is a significant step forward because it reveals the roots of the polynomial, which are the values of $x$ that make the polynomial equal to zero. These roots are the critical points that will help us solve the inequality.

Once we have factored the polynomial inequality $x(x - 1)(x - 3) gtr 0$, the next important step is to identify the critical points. Critical points are the values of $x$ for which the polynomial equals zero. These points divide the number line into intervals, and within each interval, the polynomial will either be consistently positive or consistently negative. Finding these critical points is essential for determining the solution set of the inequality.

To find the critical points, we set each factor of the polynomial equal to zero and solve for $x$:

1. $x = 0$
2. $x - 1 = 0 gtr x = 1$
3. $x - 3 = 0 gtr x = 3$

So, the critical points are $x = 0$, $x = 1$, and $x = 3$. These points are the key to understanding the behavior of the polynomial across the number line. They represent the values of $x$ where the polynomial changes its sign, transitioning from positive to negative or vice versa.

These critical points divide the number line into four intervals: $(-\infty, 0)$, $(0, 1)$, $(1, 3)$, and $(3, \infty)$. To solve the inequality, we need to determine the sign of the polynomial within each of these intervals. This is typically done by choosing a test value within each interval and evaluating the polynomial at that value. The sign of the polynomial at the test value will indicate the sign of the polynomial throughout the entire interval.

After identifying the critical points and dividing the number line into intervals, the next crucial step in solving the inequality $x(x - 1)(x - 3) gtr 0$ is to test each interval. Testing intervals involves selecting a test value within each interval and evaluating the polynomial at that value. The sign of the polynomial at the test value will indicate the sign of the polynomial throughout the entire interval. This process allows us to determine where the polynomial is positive, negative, or zero, which is essential for finding the solution set of the inequality.

We have four intervals to consider: $(-\infty, 0)$, $(0, 1)$, $(1, 3)$, and $(3, \infty)$. Let's choose a test value within each interval and evaluate the polynomial $x(x - 1)(x - 3)$:

1. Interval $(-\infty, 0)$: Choose $x = -1$

   • Polynomial value: $(-1)(-1 - 1)(-1 - 3) = (-1)(-2)(-4) = -8$. The polynomial is negative in this interval.

2. Interval $(0, 1)$: Choose $x = 0.5$

   • Polynomial value: $(0.5)(0.5 - 1)(0.5 - 3) = (0.5)(-0.5)(-2.5) = 0.625$. The polynomial is positive in this interval.

3. Interval $(1, 3)$: Choose $x = 2$

   • Polynomial value: $(2)(2 - 1)(2 - 3) = (2)(1)(-1) = -2$. The polynomial is negative in this interval.

4. Interval $(3, \infty)$: Choose $x = 4$

   • Polynomial value: $(4)(4 - 1)(4 - 3) = (4)(3)(1) = 12$. The polynomial is positive in this interval.

By testing these intervals, we have determined the sign of the polynomial in each interval. This information is crucial for identifying the solution set of the inequality.

Having tested the intervals and determined the sign of the polynomial $x(x - 1)(x - 3)$ in each interval, we can now determine the solution set for the inequality $x(x - 1)(x - 3) gtr 0$. The inequality asks for the values of $x$ where the polynomial is less than or equal to zero. This means we are looking for the intervals where the polynomial is negative or zero.

From our interval testing, we found that:

• The polynomial is negative in the intervals $(-\infty, 0)$ and $(1, 3)$.
• The polynomial is zero at the critical points $x = 0$, $x = 1$, and $x = 3$.
• The polynomial is positive in the intervals $(0, 1)$ and $(3, \infty)$.

Since we are looking for where the polynomial is less than or equal to zero, we include the intervals where the polynomial is negative and the points where it is equal to zero. Therefore, the solution set consists of the intervals $(-\infty, 0]$ and $[1, 3]$. We use square brackets to indicate that the endpoints (0, 1, and 3) are included in the solution set because the inequality includes the "equal to" condition.

In interval notation, the solution set is expressed as:

$(-\infty, 0] \cup [1, 3]$

This notation indicates that the solution set is the union of the two intervals, including the endpoints. This means that any value of $x$ within these intervals will satisfy the inequality $x^3 - 4x^2 + 3x gtr 0$.

In conclusion, we have successfully solved the polynomial inequality $x^3 - 4x^2 + 3x gtr 0$ by following a systematic approach. This approach involved factoring the polynomial, identifying the critical points, testing intervals, and determining the solution set. We began by factoring the polynomial into $x(x - 1)(x - 3)$, which allowed us to easily identify the critical points: $x = 0$, $x = 1$, and $x = 3$. These critical points divided the number line into four intervals, and we tested each interval to determine the sign of the polynomial.

By evaluating the polynomial at test values within each interval, we found that the polynomial is negative in the intervals $(-\infty, 0)$ and $(1, 3)$, positive in the intervals $(0, 1)$ and $(3, \infty)$, and zero at the critical points. Since the inequality required the polynomial to be less than or equal to zero, we included the intervals where the polynomial was negative and the critical points where it was zero. This led us to the solution set $(-\infty, 0] \cup [1, 3]$.

Understanding how to solve polynomial inequalities is a fundamental skill in algebra and calculus. The techniques discussed in this article, such as factoring, finding critical points, and testing intervals, are applicable to a wide range of polynomial inequalities. By mastering these techniques, you can confidently solve similar problems and apply this knowledge to more advanced mathematical concepts. This step-by-step approach not only provides the solution to the specific inequality but also enhances your problem-solving skills in mathematics.