# Lesson 6Sorry, We’re ClosedDevelop Understanding

### 1.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

### 2.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

### 3.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

#### d.

Find the minimum value of .

### 4.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

#### d.

Find the minimum or maximum value of . Indicate whether it’s a minimum or a maximum.

## Set

### 5.

Give an argument for the statement: The sum of the integers and is always odd. (Consider the case where one of the given integers is negative.) Then write a statement about the sum of even and odd integers.

### 6.

Write the polynomials that are analogous to the integers and .

### 7.

Given two polynomials and .

The constant term in is even and the constant term of is odd. Would it be correct to assert that the constant term of would be odd? Explain.

Identify the following statement as sometimes true, always true, or never true.

• If your answer is sometimes true, give an example of when it’s true and an example of when it’s not true.

• If it’s never true, give a counterexample.

### 8.

The set of linear polynomials is closed under multiplication.

### 9.

The set of all polynomials is closed under multiplication.

### 10.

The set of cubic polynomials is closed under subtraction.

### 11.

The set of all polynomials is closed under subtraction.

### 12.

The set of all polynomials is closed under division.
The text passage contains a series of mathematical exercises and questions, each with multiple parts, but no answers are provided within the text.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Lesson 6Sorry, We’re ClosedDevelop Understanding

### 1.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

### 2.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

### 3.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

#### d.

Find the minimum value of .

### 4.

Graph .

#### b.

Use the graph to find when .

#### c.

Use the graph to find the value of .

#### d.

Find the minimum or maximum value of . Indicate whether it’s a minimum or a maximum.

## Set

### 5.

Give an argument for the statement: The sum of the integers and is always odd. (Consider the case where one of the given integers is negative.) Then write a statement about the sum of even and odd integers.

### 6.

Write the polynomials that are analogous to the integers and .

### 7.

Given two polynomials and .

The constant term in is even and the constant term of is odd. Would it be correct to assert that the constant term of would be odd? Explain.

Identify the following statement as sometimes true, always true, or never true.

• If your answer is sometimes true, give an example of when it’s true and an example of when it’s not true.

• If it’s never true, give a counterexample.

### 8.

The set of linear polynomials is closed under multiplication.

### 9.

The set of all polynomials is closed under multiplication.

### 10.

The set of cubic polynomials is closed under subtraction.

### 11.

The set of all polynomials is closed under subtraction.

### 12.

The set of all polynomials is closed under division.
user
The text passage contains a series of mathematical exercises and questions, each with multiple parts, but no answers are provided within the text.

TYPE: MULTIPLE_QUESTIONS