MHB How do we solve a quadratic inequality with multiple factors?

[mathdad]

This is the last quadratic inequality problem (for now) before moving on to Chapter 3, Section 3.1 THE DEFINITION OF A FUNCTION.

Section 2.6
Question 30

Solve the quadratic inequality.

(x^4)(x - 2)(x - 16) ≥ 0

Do I set each factor to 0 and solve for x? The values of x are then plotted on the number line for testing.

Correct?

 

[MarkFL]

This is the last quadratic inequality problem (for now) before moving on to Chapter 3, Section 3.1 THE DEFINITION OF A FUNCTION.

Section 2.6
Question 30

Solve the quadratic inequality.

(x^4)(x - 2)(x - 16) ≥ 0

Do I set each factor to 0 and solve for x? The values of x are then plotted on the number line for testing.

Correct?

Yes, observe that the root $x=0$ is of even multiplicity (4), and so the sign of the expression will not change across this root. The others are of odd multiplicity (1) and so the sign will change across those roots.

 

[mathdad]

I will work on this tomorrow. Working right now.

 

[mathdad]

(x^4)(x - 2)(x - 16) ≥ 0

Our critical points are x = 0, x = 2 and x = 16.

I can see that our critical points are also included.

<-------(0)--------(2)--------(16)------>

For (-infinity, 0), let x = -1. True statement.

For (0, 2), let x = 1. True statement.

For (2, 16), let x = 3. False statement.

For (16, infinity), let x = 4. True statement.

Solution:

(-infinity, 2] U [16, infinity)

Correct?