Critical Numbers and Polynomial Inequalities

The video tutorial covers section 2.7 on non-linear inequalities, focusing on solving polynomial inequalities by identifying critical numbers and setting up test intervals. It includes two examples: solving a quadratic inequality and a cubic inequality using factorization. The tutorial emphasizes understanding polynomial graphs, sign changes at zeros, and the importance of arranging critical numbers to determine test intervals. The session concludes with a call to practice more problems.

What is the main focus of Section 2.7 in Chapter 2?

• Linear equations

• Non-linear inequalities

• Rational functions

• Quadratic equations

When graphing a polynomial, what are we looking for when solving inequalities?

• Regions above the x-axis

• Regions below the x-axis

• Points of intersection with the y-axis

• Maximum and minimum points

What happens at the zeros of a polynomial function?

• The function reaches its maximum value

• The function becomes undefined

• The function changes signs

• The function has a horizontal tangent

What are critical numbers in the context of polynomials?

• Zeros of the function

• Numbers where the function has a maximum

• Numbers where the function is undefined

• Numbers where the function is continuous

How do we determine test intervals for a polynomial inequality?

• By identifying the zeros and arranging them in order

• By finding the points of inflection

• By calculating the derivative

• By finding the maximum and minimum points

In Example 1, what are the critical numbers for the inequality x^2 - 5x - 6 < 0?

• 1 and -6

• 0 and 5

• 2 and -3

• -1 and 6

What is the purpose of setting up test intervals in solving inequalities?

• To find the points of inflection

• To calculate the derivative

• To determine the sign of the function in each interval

• To find the maximum value of the function

In Example 2, what is the first step in solving the inequality 3x^3 - 4x^2 - 12x > 16?

• Calculating the maximum value

• Finding the derivative

• Rewriting the inequality to have zero on one side

• Graphing the function

What method is used to factor the expression in Example 2?

• Completing the square

• Grouping

• Using the quadratic formula

• Synthetic division

What are the critical numbers found in Example 2?

• -2, 4/3, and 2

• 2, 3, and 4

• 0, 1, and 3

• -1, 0, and 1