We appreciate your visit to Solve the inequality tex 4x 3 20x 2 9x 45 textgreater 0 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve the inequality [tex]\(4x^3 + 20x^2 - 9x - 45 > 0\)[/tex], we can approach it by analyzing the polynomial function and using intervals where the function is positive. Here's how:
1. Identify the Signs of the Polynomial:
- We need to find the intervals where the polynomial [tex]\(4x^3 + 20x^2 - 9x - 45\)[/tex] is greater than zero.
2. Find the Critical Points (Roots):
- To do this, we first find the roots of the equation [tex]\(4x^3 + 20x^2 - 9x - 45 = 0\)[/tex]. These roots divide the number line into intervals that we can test.
- Assume we find the roots to be [tex]\(a\)[/tex], [tex]\(b\)[/tex], and potentially [tex]\(c\)[/tex].
3. Test the Intervals:
- The roots divide the number line into intervals. Typically, we check the sign of the polynomial in each interval.
- The intervals we are interested in come from considering the sorted roots and checking between and beyond them.
4. Determine Where the Polynomial is Positive:
- Analyze the intervals: if we assume the roots found are [tex]\(-5\)[/tex], [tex]\(-\frac{3}{2}\)[/tex], and [tex]\(\frac{3}{2}\)[/tex], then the intervals are:
- [tex]\((-5, -\frac{3}{2})\)[/tex]
- [tex]\((-\frac{3}{2}, \frac{3}{2})\)[/tex]
- [tex]\((\frac{3}{2}, \infty)\)[/tex]
- Evaluate the sign of the polynomial in each interval.
5. Construct the Solution:
- Through your interval analysis, you'll find that the polynomial [tex]\(4x^3 + 20x^2 - 9x - 45\)[/tex] is positive in:
- The interval [tex]\((-5, -\frac{3}{2})\)[/tex]
- The interval [tex]\((\frac{3}{2}, \infty)\)[/tex]
Therefore, the solution to the inequality [tex]\(4x^3 + 20x^2 - 9x - 45 > 0\)[/tex] is:
- For [tex]\(x\)[/tex] in the interval [tex]\((-5, -\frac{3}{2})\)[/tex].
- For [tex]\(x\)[/tex] in the interval [tex]\((\frac{3}{2}, \infty)\)[/tex].
This means the polynomial is positive in these regions of [tex]\(x\)[/tex], and any [tex]\(x\)[/tex] values within these intervals will satisfy the given inequality.
1. Identify the Signs of the Polynomial:
- We need to find the intervals where the polynomial [tex]\(4x^3 + 20x^2 - 9x - 45\)[/tex] is greater than zero.
2. Find the Critical Points (Roots):
- To do this, we first find the roots of the equation [tex]\(4x^3 + 20x^2 - 9x - 45 = 0\)[/tex]. These roots divide the number line into intervals that we can test.
- Assume we find the roots to be [tex]\(a\)[/tex], [tex]\(b\)[/tex], and potentially [tex]\(c\)[/tex].
3. Test the Intervals:
- The roots divide the number line into intervals. Typically, we check the sign of the polynomial in each interval.
- The intervals we are interested in come from considering the sorted roots and checking between and beyond them.
4. Determine Where the Polynomial is Positive:
- Analyze the intervals: if we assume the roots found are [tex]\(-5\)[/tex], [tex]\(-\frac{3}{2}\)[/tex], and [tex]\(\frac{3}{2}\)[/tex], then the intervals are:
- [tex]\((-5, -\frac{3}{2})\)[/tex]
- [tex]\((-\frac{3}{2}, \frac{3}{2})\)[/tex]
- [tex]\((\frac{3}{2}, \infty)\)[/tex]
- Evaluate the sign of the polynomial in each interval.
5. Construct the Solution:
- Through your interval analysis, you'll find that the polynomial [tex]\(4x^3 + 20x^2 - 9x - 45\)[/tex] is positive in:
- The interval [tex]\((-5, -\frac{3}{2})\)[/tex]
- The interval [tex]\((\frac{3}{2}, \infty)\)[/tex]
Therefore, the solution to the inequality [tex]\(4x^3 + 20x^2 - 9x - 45 > 0\)[/tex] is:
- For [tex]\(x\)[/tex] in the interval [tex]\((-5, -\frac{3}{2})\)[/tex].
- For [tex]\(x\)[/tex] in the interval [tex]\((\frac{3}{2}, \infty)\)[/tex].
This means the polynomial is positive in these regions of [tex]\(x\)[/tex], and any [tex]\(x\)[/tex] values within these intervals will satisfy the given inequality.
Thanks for taking the time to read Solve the inequality tex 4x 3 20x 2 9x 45 textgreater 0 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
