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Polynomial Inequalities. The last section in Chapter 2!!! Boo Hoo  2.9a. Polynomial Inequalities. They can be written in the forms:. To solve the inequality f ( x ) > 0 is to find the values of x that make f ( x ) positive.

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Polynomial Inequalities


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polynomial inequalities

Polynomial Inequalities

The last section in Chapter 2!!!

Boo Hoo 2.9a

polynomial inequalities1
Polynomial Inequalities

They can be written in the forms:

  • To solve the inequality f (x) > 0 is to find the values of x that
  • make f (x) positive
  • To solve the inequality f (x) < 0 is to find the values of x that
  • make f (x) negative

If the expression f (x) is a product, we can find its sign by looking

at the sign of each of its factors…

polynomial inequalities2
Polynomial Inequalities

Determine the real number values of x that cause the given

function to be (a) zero, (b) positive, (c) negative.

Real zeros: x = –3, 4

We can use a sign chart to find the other values of x:

Negative

Positive

Positive

–3

4

The function is positive on

The function is negative on

polynomial inequalities3
Polynomial Inequalities

Determine the real number values of x that cause the given

function to be (a) zero, (b) positive, (c) negative.

Negative

Positive

Positive

–3

4

The function is positive on

The function is negative on

Does this make sense given the multiplicity of the zeros?

Can we verify all of this graphically ?

polynomial inequalities4
Polynomial Inequalities

For the same function, give the solution of each of the following:

polynomial inequalities5
Polynomial Inequalities

Solve the given inequality analytically.

Let

Use the RZT to list the possible rational zeros!!!

Look at the graph  x = 4 seems like a good guess…

polynomial inequalities6
Polynomial Inequalities

Solve the given inequality analytically.

4

2

–7

–10

24

8

4

–24

2

1

–6

0

polynomial inequalities7
Polynomial Inequalities

Solve the given inequality analytically.

Zeros: 4, 3/2, –2  all of multiplicity 1!!!

Give the end behavior of this function!!!

polynomial inequalities8
Polynomial Inequalities

Solve the given inequality analytically.

Solution:

Now, for the sign chart:

Sign

Change

Sign

Change

Sign

Change

Neg. due

to end

behavior

Positive

Negative

Pos. due

to end

behavior

x

–2

3/2

4

polynomial inequalities9
Polynomial Inequalities

Solve the given inequality graphically.

Set equal to zero:

Rewrite as a function:

Graph to find the real zeros and solve the inequality!!!

Solution

polynomial inequalities with unusual answers
Polynomial Inequalities with Unusual Answers…

Give the solution to each of the following:

The graph???

No Solution

No Solution

polynomial inequalities with unusual answers1
Polynomial Inequalities with Unusual Answers…

Give the solution to each of the following:

The graph???

No Solution