Simplifying Rational Expressions

Simplifying a Fraction Simplify: The techniques we use to simplify a fraction without variables (Finding the greatest common FACTOR) is the same we will use to simplify fractions with variables.

Rational Expressions A Rational Expression is an algebraic fraction: a fraction that contains a variable(s). Our goal is to simplifying rational expressions by “canceling” off common factors between the numerator and denominator. Similar to simplifying a numeric fraction. Example: We will see how to simplify the original expression.

Simplifying Rational Expressions Simplify the following expressions by finding a common factor: Can we simplify this one more?

The Major Requirement for Simplifying Rational Expressions A fellow student simplifies the following expressions: Which simplification is correct? Substitute two values of x into each to justify your answer. MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator are single terms and are product of factors. Equal. Not Equal.

Which is Simplified Correctly? Which of the following expressions is simplified correctly? Explain how you know. MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator are single terms and are product of factors. XLeftRight XLeftRight The left side of the equation has to equal the right. Make a table for each side of the equation to see if they are the same.

Back to a Previous Example… Can we simplify the following expression more? NO!! IT MUST BE MUITLIPLICATION! The numerator is the sum of two terms. The denominator is the difference of two terms. The numerator and the denominator are not written as a product. Also, there greatest common factor is 1. If you still disagree, make a table to check your hypothesis.

Example 1 State the values that make the denominator zero and then simplify: Half the work is done. It is already factored. CAN cancel since the top and bottom have common factors. Rewrite any factors if they are raised to a power Don’t forget about cancelling common numeric Factors. Make the Denominator 0: 2 and -7. These Make the ORIGINAL denominator equal 0. We assume that x can never be these values.

Example 2 Can NOT cancel since its not in factored form. Also it is not obvious what values of x make the denominator 0. CAN cancel since the top and bottom have a common factor Always Factor Completely 4, -4, and 0. These Make the ORIGINAL denominator equal 0. We assume that x can never be these values. State the values that make the denominator zero and then simplify: Make the Denominator 0: The denominator is factored, so it is obvious what values of x make it 0

Example 3 Can NOT cancel any factors since its not in factored form CAN cancel common factors since the top and bottom have a common factor If they are not quadratics, find a common factor. State the values that make the fraction undefined and then simplify: a=0 or b=0 These Make the ORIGINAL denominator equal 0. We assume that a & b can never be these values. Make the Fraction Undefined: Finding where the fraction undefined is the same as finding when the denominator is 0.