Rational Expressions and Polynomial Division

The video tutorial explores the division of simple polynomials and examines whether polynomials are closed under division. It begins with a review of sets and polynomials, explaining that rational expressions are formed by dividing polynomials. The tutorial then demonstrates how to simplify rational expressions by factoring and addresses common misunderstandings in polynomial division. Through examples, it illustrates the process of dividing polynomials and concludes with a discussion on closure under division, providing a counterexample to show that polynomials are not always closed under division.

What is a rational expression?

• An expression that cannot be simplified

• An expression with only positive numbers

• An expression that can be written as a fraction of two polynomials

• An expression with only whole numbers

Which of the following is a numeric set?

• Set of whole numbers

• Set of quadrilaterals

• Set of letters

• Set of colors

What is the key step in simplifying a rational expression?

• Subtracting the numerator from the denominator

• Factoring the numerator and denominator

• Multiplying the numerator and denominator

• Adding the numerator and denominator

When simplifying rational expressions, what should you avoid doing?

• Checking for common factors

• Dividing equivalent factors

• Crossing out terms that are not equivalent factors

• Factoring both the numerator and denominator

In the expression 35x^7 - 10x^5 - 25x^4 divided by 5x^2, what is the first step?

• Subtract 5x^2 from each term

• Multiply the numerator by 5x^2

• Factor out 5x^2 from the numerator

• Add 5x^2 to each term

What is the result of dividing b^2 - 9 by b + 3?

• b + 3

• b + 9

• b - 3

• b^2 - 3

What does it mean for polynomials to be closed under division?

• Polynomials cannot be divided

• The quotient of any two polynomials is always a rational number

• Polynomials can only be divided by integers

• The quotient of any two polynomials is always a polynomial

What is a counterexample in the context of polynomial division?

• An example that is irrelevant

• An example that disproves a hypothesis

• An example that is incomplete

• An example that supports a hypothesis

In the example 4m^5 - 18m^3 + 24m divided by 4m^2, what is the resulting expression?

• m^3 - 9m - 6

• m^3 + 9m - 6

• m^3 - 9/2m + 6m^-1

• m^3 - 9m + 6

Why is the quotient m^3 - 9/2m + 6m^-1 not a polynomial?

• It has a zero coefficient

• It has a positive exponent

• It has a fractional coefficient

• It has a negative exponent