Lesson 3-2 Polynomial Inequalities in One Variable

Objective:

Objective: To solve and graph polynomial inequalities in one variable.

Polynomial Inequalities:

P(x) > 0 or P(x) < 0

Polynomial Inequalities: P(x) > 0 or P(x) < 0 a) Use a sign graph of P(x).

Polynomial Inequalities: P(x) > 0 or P(x) < 0 b) Analyze a graph of P(x).

Polynomial Inequalities: P(x) > 0 or P(x) < 0 b) Analyze a graph of P(x). i. P(x) > 0 when the graph is above the x-axis.

Polynomial Inequalities: P(x) > 0 or P(x) < 0 b) Analyze a graph of P(x). i. P(x) > 0 when the graph is above the x-axis. ii. P(x) < 0 when the graph is below the x-axis.

Example:

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph.

Example: Step 1: Find the zeros of the polynomial.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 1: Find the zeros of the polynomial. P(x) = x 3 – 2x 2 – 3x P(x) = x(x 2 – 2x – 3) P(x) = x(x – 3)(x + 1) Zeros: x = 0, x = 3, x = - 1

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 Now, these 3 zeros separate this number line into 4 areas.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 All the values less than -1, all the values between -1 and 0, all the values between 0 and 3, and all the values greater than 3.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 Now, pick a number in an area, like 1. Substitute 1 in for x. 1(1-3)(1+1)  1(-2)(2)  -4

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 Now, pick a number in an area, like 1. Substitute 1 in for x. 1(1-3)(1+1)  1(-2)(2)  -4

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 So, that means when x = 1, P(x), which is y, is negative.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 Now, using our rule for roots we discovered a chapter ago, the graph of P(x) will change signs after it goes through each root, since they are all single roots.

Example: Solve x 3 – 2x 2 – 3x < 0 by using a sign graph. Step 2: Plot the zeros on a number line. 03 Using this sign graph and the fact that we are dealing with P(x) < 0 (which means negative numbers) the answer to the problem is: x< -1 U 0 < x < 3 or (- 8, -1) U (0,3)

Example: Solve (x 2 – 1)(x – 4) 2 > 0.

Example:

Example:

Assignment: Pg. 103 C.E.  1-6 all W.E.  1-19 odd