Operations and Properties of Polynomials

This video tutorial explains which operations are closed for polynomials by applying the definition of a polynomial to each operation. It reviews closed operations for integers, highlighting that they are closed under addition, subtraction, and multiplication but not division. The video also addresses common misunderstandings about natural numbers and their operations. Finally, it demonstrates closed operations for polynomials, showing that addition, subtraction, and multiplication result in polynomials, while division does not always yield a polynomial.

What operations are integers closed under?

• Multiplication and division

• Addition, subtraction, and multiplication

• Addition, subtraction, and division

• Subtraction and division

Why are integers not closed under division?

• Because division always results in a fraction

• Because division sometimes results in a non-integer

• Because division always results in an integer

• Because division is not defined for integers

Which operation is not closed for natural numbers?

• Subtraction

• All operations are closed

• Addition

• Multiplication

What is a common misunderstanding about closed operations?

• Multiplication is not closed for integers

• Subtraction is closed for integers

• All operations are closed for natural numbers

• If one example works, the operation is closed

What must terms in a polynomial be separated by?

• Parentheses

• Exponents

• Addition or subtraction

• Multiplication or division

What is the result of adding two polynomials?

• A fraction

• A polynomial

• An integer

• A non-polynomial expression

What is the result of multiplying two polynomials?

• A non-polynomial expression

• An integer

• A polynomial

• A fraction

What happens when you divide two polynomials?

• The result is never a polynomial

• The result may not be a polynomial

• The result is always a polynomial

• The result is always an integer

Why is division not a closed operation for polynomials?

• Because division always results in a polynomial

• Because division sometimes results in a non-polynomial

• Because division is not defined for polynomials

• Because division always results in a fraction

What did you learn about polynomial operations in this lesson?

• Only division is closed for polynomials

• Only addition and multiplication are closed for polynomials

• All operations are closed for polynomials

• Addition, subtraction, and multiplication are closed, but not division