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Solution of The Following Polynomial Inequality Calculator

Reviewed by Calculator Editorial Team

This calculator helps you solve polynomial inequalities by finding the values of x that satisfy the given inequality. Whether you're a student studying algebra or a professional working with mathematical expressions, this tool provides a clear solution set for inequalities involving polynomials.

Introduction

Polynomial inequalities are mathematical expressions that compare a polynomial to a value, typically zero. Solving these inequalities involves finding the range of x values that satisfy the inequality. This calculator simplifies the process by providing step-by-step solutions.

The general form of a polynomial inequality is:

P(x) > 0 P(x) ≥ 0 P(x) < 0 P(x) ≤ 0

where P(x) is a polynomial in x. The solution involves finding the roots of P(x) and determining the intervals where the inequality holds true.

How to Use the Calculator

Using the calculator is straightforward:

  1. Enter the polynomial inequality in the input field. For example, type "x² - 5x + 6 ≤ 0".
  2. Click the "Calculate" button to solve the inequality.
  3. Review the solution set displayed in the result panel.
  4. Optionally, view the graphical representation of the polynomial.

The calculator will provide the solution set in interval notation and explain the steps taken to arrive at the answer.

Methodology

The calculator uses the following steps to solve polynomial inequalities:

  1. Find the roots of the polynomial P(x) = 0.
  2. Plot the roots on a number line.
  3. Test the intervals between the roots to determine where P(x) is positive or negative.
  4. Combine the intervals that satisfy the original inequality.

For example, solving x² - 5x + 6 ≤ 0 involves finding the roots of x² - 5x + 6 = 0, which are x = 2 and x = 3. The solution set is then determined by testing the intervals.

Examples

Here are a few examples of polynomial inequalities and their solutions:

Example 1

Inequality: x² - 4x + 3 > 0

Solution: x < 1 or x > 3

Example 2

Inequality: x³ - 2x² - x + 2 ≤ 0

Solution: -1 ≤ x ≤ 1 or x = 2

Example 3

Inequality: 2x² + 3x - 2 < 0

Solution: -2 < x < 0.5

These examples demonstrate how the calculator can handle different types of polynomial inequalities.

FAQ

What types of polynomial inequalities can this calculator solve?

This calculator can solve inequalities of the form P(x) > 0, P(x) ≥ 0, P(x) < 0, and P(x) ≤ 0, where P(x) is a polynomial in x.

How accurate are the solutions provided by the calculator?

The calculator uses precise mathematical methods to find the solution set. However, it's always good practice to verify the results, especially for complex polynomials.

Can the calculator handle inequalities with fractional coefficients?

Yes, the calculator can handle polynomials with fractional coefficients. Simply enter the inequality as you would write it mathematically.