Problem 81 - Algebra and Trigonometry

Question

What is a polynomial inequality?

Step-by-Step Solution

Answer

A polynomial inequality is a type of inequality that involves a polynomial. It is solved by finding the roots of the polynomial, dividing the number line into intervals, and substituting values from each interval into the equation to determine the solution set.

Step 1: Definition

A Polynomial Inequality is an inequality that involves a polynomial. Basically, it's an inequality which includes a polynomial expression. Instead of being equal to each other, the expressions are either greater than, less than, or not equal to each other.

Step 2: Explanation of the Parts

The simplest type of polynomial inequality has the form \(P(x) > 0\), \(P(x) < 0\), \(P(x) ≥ 0\), or \(P(x) ≤ 0\), where \(P(x)\) is a polynomial function.

Step 3: Solving

To find the solution of the inequality, we find the roots of the polynomial and use these to define intervals on the number line. Substituting a value from each interval into the polynomial provides the sign of the polynomial in that interval, which allows us to solve the inequality.