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- 1. 3-2 Polynomial Inequalities in One Variable Objective: Solve polynomial inequalities in one variable by: 1. Using a sign graph 2. Analyzing a graph of P(x)
- 2. 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 graph 2. Analyzing a graph of P(x)
- 3. 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
- 4. Example 1 • Solve x3 – 2x2 – 3x < 0 using a sign graph.
- 5. You Try! • Solve: 2x2 + 3x – 5 < 0
- 6. Example 2 • Solve (x2 – 1)(x – 4)2 0
- 7. You Try! • Solve: x4 – 4x2 0
- 8. 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!
- 9. Example 3 • Solve
- 10. You Try! • Solve
- 11. 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
- 12. Example 4 • Solve 2x3 + x2 – 8x + 3 > 0 using a graphing calculator.
- 13. You Try! • Solve 4x3 – 3x2 – 9x – 2 0

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