3-2 Polynomial Inequalities in One VariableObjective:Solve polynomial inequalities in one variable by:1. Using a sign graph2. Analyzing a graph of P(x)

Polynomial Inequalities•If P(x) is a polynomial, then P(x) > 0 and

P(x) < 0 are polynomial inequalities.

•We will learn two methods to solve:1. Using a sign graph2. Analyzing a graph of P(x)

Method 1 – Using a Sign Graph

•Use if the polynomial is factorable.

•Factor to find zeros•Plot on a number line•Test values from each interval to determine

the sign of P(x)•Hint:•P(x) < 0 means find negative intervals•P(x) > 0 means find positive intervals

Example 1•Solve x3 – 2x2 – 3x < 0 using a sign graph.

You Try!•Solve: 2x2 + 3x – 5 < 0

Example 2•Solve (x2 – 1)(x – 4)2 0

You Try!•Solve: x4 – 4x2 0

Rational Inequalities •Use same method for rational inequalities

where P(x) and Q(x) are polynomials.

To solve:•Plot all zeros of numerator and denominator•Use an open dot for zeros of the denominator ▫they make the function undefined: not part of

solution•Check all intervals, don’t assume signs

alternate!

Example 3•Solve

You Try!•Solve

Method 2 – Analyze the Graph•Useful for functions that are not

factorable.

•Graph the function on the calculator.•Find zeros using trace •P(x) > 0 where graph is above x-axis•P(x) < 0 where graph is below x-axis

Example 4•Solve 2x3 + x2 – 8x + 3 > 0 using a

graphing calculator.

You Try!•Solve 4x3 – 3x2 – 9x – 2 0