Algebra Chapter 2

Questions and Answers

What is the result of dividing a negative integer by a positive integer?

The result is zero.

The result is negative. (correct)

The result is positive.

The result is undefined.

What is the quotient of (–100) ÷ 5?

–15

–20 (correct)

Which of the following represents the property of division for negative integers correctly?

Dividing a positive by a positive results in a negative.

Dividing a negative by a negative results in a negative.

Dividing a negative by a positive results in a positive.

Dividing a negative by a negative results in a positive. (correct)

If (–8) × (–9) equals 72, what is the corresponding division statement?

72 ÷ (–9) = –8 Signup and view all the answers

How do you represent (–45) ÷ 5?

It can be simplified to 9 with a minus sign. Signup and view all the answers

What is the result of dividing a negative integer by 1?

The result remains the same negative integer. Signup and view all the answers

How is division not commutative for integers demonstrated?

(–30) ÷ (–6) does not equal (–6) ÷ (–30). Signup and view all the answers

What happens when you divide any integer by (–1)?

The result is the opposite sign of the integer. Signup and view all the answers

Which statement about dividing by zero is true?

Dividing any integer by zero is not defined. Signup and view all the answers

What can you conclude from the example (–8) ÷ (–4) = 2?

Dividing two negative integers always results in a positive integer. Signup and view all the answers

Study Notes

Division of Integers

Division is not commutative for integers, meaning ( a ÷ b \neq b ÷ a ).
Example: ( (–8) ÷ (–4) = 2 ) while ( (–4) ÷ (–8) \neq 2 ).
Division of integers can yield different types of results:
- ( a ÷ 0 ) is undefined; ( 0 ÷ b = 0 ) for any ( b \neq 0 ).

Division by One

Dividing any integer by ( 1 ) results in the same integer.
- Example: ( (–8) ÷ 1 = (–8) ).
This rule holds true for negative integers as well.

Division by Negative One

Dividing an integer by ( –1 ) changes the sign of the integer.
- Example: ( (–1) × 5 = –5 ) implies ( 5 ÷ (–1) = (–5) ).

Inverse Relationship with Multiplication

Division is the inverse operation of multiplication.
- Example: For ( 3 × 5 = 15 ), it follows that ( 15 ÷ 5 = 3 ) and ( 15 ÷ 3 = 5 ).
Each multiplication statement corresponds to two division statements.

Observations on Division Results

When dividing a negative integer by a positive integer, the quotient is negative.
- Example: ( (–32) ÷ 4 = (–8) ).
Conversely, dividing a positive integer by a negative integer also results in a negative quotient.

Negative Divisions

Dividing two negative integers results in a positive quotient.
- Example: ( (–12) ÷ (–6) = 2 ).

General Division Rules

For two positive integers ( a ) and ( b ):
- ( a ÷ (–b) = (–a) ÷ b ) for ( b \neq 0 ).
- ( (–a) ÷ (–b) = a ÷ b ) for ( b \neq 0 ).

Non-closure Property

Integers are not closed under division, as not every division operation yields another integer.

Description

Test your knowledge of division with negative numbers in this algebra quiz. Analyze statements and inferences to determine the outcomes of various division operations. Perfect for students looking to reinforce their understanding of integer results.