Understanding Equations and Zero Divisors

What assumption is made when solving equations by factoring in regular algebra?

• The sum of two terms is zero.

• The quotient of two terms is zero.

• The product of two terms is zero.

• The difference of two terms is zero.

In the integers mod 12, how many solutions were found for the equation x² + 5x + 6 = 0?

• Two solutions

• Five solutions

• Three solutions

• Four solutions

What are zero divisors?

• Numbers that subtract to zero in a ring

• Numbers that add to zero in a ring

• Numbers that multiply to zero in a ring

• Numbers that divide zero in any ring

Why do the integers mod 11 not have zero divisors?

• Because 11 is a prime number

• Because 11 is an even number

• Because 11 is a composite number

• Because 11 is a perfect square

What is an integral domain?

• A ring with zero divisors

• A ring without a multiplicative identity

• A commutative ring with 1 and no zero divisors

• A non-commutative ring with zero divisors

What property allows for safe cancellation in an integral domain?

• Division property

• Cancellation property

• Multiplication property

• Addition property

What happens when you factor and cancel in a ring with zero divisors?

• You cannot solve the equation

• You always find the correct solution

• You might miss some solutions

• You find more solutions than exist

Why is the term 'integral domain' used?

• Because it is a type of function

• Because it is a type of polynomial

• Because it relates to calculus

• Because it generalizes integers for number theory

What is a domain in the context of rings?

• A ring without a multiplicative identity

• A ring with zero divisors

• A ring without zero divisors

• A ring with a multiplicative identity

What humorous observation is made about integral domains?

• They are only a function's domain

• They are both integral and a function's domain

• They are only integral

• They are neither integral nor a function's domain