Simplifying Rational Expressions: Key Concepts and Techniques

This video tutorial introduces rational expressions, explaining their definition and domain. It covers the process of simplifying rational expressions by factoring numerators and denominators and removing common factors. The video provides several examples, ranging from basic to more complex cases, illustrating the simplification process. It emphasizes the importance of ensuring the simplified expression remains defined within the original expression's domain. The tutorial concludes with final examples and a summary of key points.

What is a rational expression?

• An equation that equates two fractions

• A polynomial raised to a negative exponent

• Any algebraic expression involving a variable

• A fraction where both numerator and denominator are polynomials

What condition must be met for a rational expression to be defined?

• The numerator must be non-zero

• The denominator must be non-zero

• Both numerator and denominator must be equal

• The expression must contain at least one variable

How can a rational expression be simplified?

• By adding common factors

• By multiplying the numerator and denominator by the same number

• By factoring and cancelling out common factors

• By subtracting any terms

Why can't we simplify across addition or subtraction in rational expressions?

• Because it violates algebraic principles

• Because it can lead to incorrect results

• All of the above

• Because it changes the value of the expression

What happens if a variable in the denominator of a rational expression equals zero?

• The expression simplifies to one

• The expression is undefined

• The expression remains unchanged

• The expression equals zero

In the expression (3x^2 - 10x - 8) / (x^2 - 7x + 12), what common factor can be cancelled?

• x - 4

• x - 3

• x + 2

• 3x + 2

What is the simplified form of the expression (x^2 - 9) / (x^2 - 2x - 15)?

• (x - 3) / (x + 5)

• (x - 3) / (x - 5)

• (x + 3) / (x - 5)

• (x + 3) / (x + 5)

If the original rational expression has x - 7 in the denominator, which value of x makes it undefined?

• x = 1

• x = 7

• x = -7

• x = 0

What is the result of simplifying the expression (x - 7) / (21 - 3x)?

• -1/3

• 1/3

• -3

• 3

Which of the following is a correct step in simplifying rational expressions?

• Cancel out terms across addition

• Ignore the domain restrictions

• Factor both numerator and denominator completely

• Multiply the numerator by the denominator