Lesson 3-2

Polynomial Inequalities in One Variable

http://www.mathsisfun.com/algebra/inequality-quadratic-solving.html

Objective:Objective:

Objective:Objective:

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

Polynomial Inequalities:

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:

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 1: Find the zeros of the polynomial.

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 1: Find the zeros of the polynomial.

P(x) = x3 – 2x2 – 3xP(x) = x(x2 – 2x – 3)P(x) = x(x – 3)(x + 1)Zeros: x = 0, x = 3, x = - 1

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

Now, these 3 zeros separate this number line into 4 areas.

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

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:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

Now, pick a number in an area, like 1. Substitute 1 in for x.

1(1-3)(1+1) 1(-2)(2) -4

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

Now, pick a number in an area, like 1. Substitute 1 in for x.

1(1-3)(1+1) 1(-2)(2) -4

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

So, that means when x = 1, P(x), which is y, is negative.

Example:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1

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:Example:Solve x3 – 2x2 – 3x < 0 by using a sign graph.

Step 2: Plot the zeros on a number line.

0 3-1Using 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:Example:Solve (x2 – 1)(x – 4)2 > 0.

Example:Example:

Example:Example:

Assignment:Assignment:

Pg. 103C.E. 1-6 all

W.E. 1-19 odd