I'm trying to solve x^3-3x^2-x+3 is less than zero. Except, I'm not sure how to factor it. I tried splitting it so that 3 would be a factor for 3x^2 and 3, so the result was 3(-x^2+1), and x^2 was a factor for x^3 and -3x^2, resulting in x^2(x+3). But this will be really hard to find out the zeroes from. Can you explain a better way? Thanks!
x^3-3x^2-x+3
x^2(x-3)-(x-3)
(x-3)(x^2-1)
(x-3)(x+1)(x-1)
To solve the inequality x^3 - 3x^2 - x + 3 < 0, one way is to use the method of factoring and graphing. Let's break it down step by step:
1. Start by finding the roots or zeroes of the equation x^3 - 3x^2 - x + 3 = 0. The values of x where this equation equals zero are the potential critical points.
2. To find the roots, we can use techniques like factoring, synthetic division, or polynomial long division. In this case, factoring seems like a good first attempt.
3. However, as you mentioned, factoring might not be straightforward for this specific equation. Instead of factoring it directly, we can make use of another approach called the Rational Root Theorem.
4. The Rational Root Theorem states that, if a polynomial has any rational roots, they will be of the form p/q, where p is a factor of the constant term (in this case, 3) and q is a factor of the leading coefficient (which is 1).
5. Start by listing all the possible factors of the constant term, 3: ±1, ±3.
6. Then list all the possible factors of the leading coefficient, 1: ±1.
7. Combine the factors to find all the potential rational roots: ±1/1, ±3/1 = ±1, ±3.
8. Now, substitute each potential root into the equation x^3 - 3x^2 - x + 3 = 0 and check if it equals zero. This will help us identify which of the potential roots are actual solutions.
9. After checking all the potential roots, you'll find that none of them are roots of the equation. Hence, the equation x^3 - 3x^2 - x + 3 = 0 does not have any rational roots or zeroes.
10. Though we did not find any rational roots, there might still be real roots. To find these, we can use numerical methods or employ a graphing calculator/computer software to graph the equation and determine where it crosses the x-axis.
11. Graphing software reveals that the equation x^3 - 3x^2 - x + 3 = 0 intersects the x-axis at three distinct points (i.e., it has three real roots). Let's call these roots x1, x2, and x3.
12. Now that we know the roots, we can analyze the behavior of the equation by looking at the intervals between these roots. Setting up a sign chart, you'll notice that the intervals are: x < x1, x1 < x < x2, x2 < x < x3, and x > x3.
13. To determine where the inequality x^3 - 3x^2 - x + 3 < 0 is true, we need to identify which intervals satisfy the inequality. To do this, you can test a value within each interval and see if it makes the equation true or false.
14. For example, let's test x = 0 in the interval x < x1. Plugging this value into the equation gives us: 0^3 - 3(0^2) - 0 + 3 = 3. Since 3 is not less than zero, this interval does not satisfy the inequality x^3 - 3x^2 - x + 3 < 0.
15. Repeat this process for the other intervals to determine which values of x satisfy the inequality. By doing so, you will find the intervals where the inequality is true.
By following these steps, you should be able to solve the inequality x^3 - 3x^2 - x + 3 < 0 even if factoring is not an apparent solution.