Polynomial Inequalities in One Variable Section 3-2 Polynomial Inequalities in One Variable

Polynomial Inequalities Let P(x) be any polynomial. Then P(x) < 0 and P(x) > 0 are called polynomial inequalities. There are two ways to solve a polynomial inequality.

Two ways to solve a Polynomial Inequality Method 1: Use a sign graph of P(x) Using a sign graph is an easy way to solve a polynomial inequality if the polynomial is factorable.

Using a Sign Analysis to Solve Polynomial Inequalities Recall… To perform a sign analysis of a polynomial P(x), you test one value of x from each of the intervals determined by the zeros of P(x). Then you determine the sign of P(x) in each of these intervals.

Two ways to solve a Polynomial Inequality Method 2: Analyze a graph of P(x). Note that P(x) > 0 when the graph is ABOVE the x-axis and P(x) < 0 when the graph is BELOW the x-axis

Using the Graphing Calculators to Solve Polynomial Inequalities Put a plus sign where the graph is above the x axis Put a minus sign where the graph is below the x axis If P(x)>0 you want plus signs If P(x)<0 you want minus signs

Example Solve using Sign Analysis Solve Using the Graphing Calculator

Activity Complete Activity on page 101.

Effect of a Squared Factor This activity shows that not all polynomials change sign at a zero. A polynomial P(x) will NOT change sign at a zero c if c corresponds to the squared factor (x – c)2.

Example Solve by Sign Analysis Solve by using the graphing calculator

Solving Polynomial Inequalities You must use the sign analysis when you have a polynomial inequality that is a fraction. See example 3 p. 102

Examples Solve: