1. Topic 1: Simplifying Rational Expressions

2. I can simplify a rational expression.
I can explain why the non-permissible values of a given rational expression and its simplified form are the same.
I can identify and correct errors in a given simplification of a rational expression, and explain the reasoning.

3. Explore…
1. Consider the rational numbers 4 6 and Are they equivalent? Explain. 2. Consider the rational expressions 4𝑥+2 6 and 2𝑥 Are they equivalent? How can we tell?

4. You should notice…
You should recognize that the process for determining equivalent rational numbers or expressions is multiplying or dividing the numerator and denominator of the fraction by the same number or expression. You should also note that when dealing with rational expressions, there may be certain values of x that would make the expression undefined (that would make the denominator = 0).

5. Information
A rational number is a fraction that has only integers for both the numerator and denominator. Division by 0 is undefined, so the denominator cannot be 0.
A rational expression is a fraction that has polynomials in both the numerator and denominator.
A non-permissible value is a value for the variable that makes the denominator of a rational expression equal to 0. A denominator of 0 makes the expression undefined.
A restriction is the statement of any non-permissible values at the end of a question along with the answer.

6. Information
A rational expression is in simplified form when the numerator and the denominator have no common factors other than 1. Non-permissible values are always stated as restrictions along with the simplified form, whether the question asks you to find them or not.
To simplify,
Factor numerator and denominator.
Find NPV’s.
Cancel factors common to numerator and denominator and state simplified expression with NPV’s.
Equivalent rational expressions are rational expressions that simplify to the same expression.

7. Remember: Any value of x that makes the denominator equal to 0 is not permissible!
Example 1
Determining the non-permissible values for a rational expression
Determine the non-permissible value(s) for each rational expression. a) b)

8. Remember: Any value of x that makes the denominator equal to 0 is not permissible!
Example 1
Determining the non-permissible values for a rational expression
Determine the non-permissible value(s) for each rational expression. c) d)
When there is a common factor, it is important to pull that out first!
This expression is equal to zero if either of the factors are equal to zero.

9. Remember to always state the non-permissible values as restrictions with the simplified form of a rational expression.
Example 2
Simplifying a rational expression with monomials
Find the GCF; then simplify the following rational expressions. a) b)
GCF: 6a2
GCF: 18x4
Remember:
The GCF is the greatest term that can be pulled out of the numerator and denominator.

10. Example 3
Create equivalent rational expressions
Create two rational expressions that are equivalent to the one given. Identify the expression that is in simplest form. a)
Can also multiply both top and bottom by the same thing to create another equivalent expression.
GCF: 5x
Simplest form

11. Example 3 b) Create equivalent rational expressions
Can also multiply both top and bottom by the same thing to create another equivalent expression.
GCF: 3y
Simplest form

12. Example 4 Simplify the following rational expressions. a)
Remember to always state the non-permissible values as restrictions with the simplified form of a rational expression.
Example 4
Simplifying a rational expression with binomials
Simplify the following rational expressions. a)
Sometimes the GCF of the numerator is different from the GCF of the denominator.
GCF (numerator): 2m GCF (denominator):m2
To find the NPV’s, use the denominator from the start of the question (in factored form).

13. Example 4 Simplify the following rational expressions. b)
Simplifying a rational expression with binomials
Simplify the following rational expressions. b)
GCF (numerator): none GCF (denominator):3
To find the NPV’s, use the denominator from the start of the question (in factored form).
Notice a-2 is exactly opposite (in signs) to 2-a. We can factor out a -1 to make them equal…

14. Example 5
Writing a rational expression
a) Write a rational expression with a binomial in the denominator, so that the expression simplifies to . State the non-permissible values.
This question can be answered infinitely many ways. One possibility is as follows:

15. Example 5
Writing a rational expression
b) Write a rational expression with a monomial in the numerator and a binomial in the denominator, so that the expression has a non-permissible value of -2. State any other non-permissible values.
This question can be answered infinitely many ways. One possibility is as follows:

16. Example 6
Finding the error
a) Max decided that the rational expressions 𝑥 2 −1 4𝑥 and 5−5𝑥 3𝑥 are equivalent by evaluating each expression for x = 1. Is Max’s solution correct? Explain.
NO. By substituting x=1 into each expression, Max shows that the rational expressions are equal for the specific value of x = 1. This does not prove equivalence.

17. Example 6
Finding the Error
b) Carrie tried to simplify the rational expression 4𝑥+8 4𝑥− Is Carrie’s solution correct? Explain.
Carrie’s solution is not correct. She cannot just cancel two terms. She can only cancel factors. She must look for GCF’s first!

18. Need to Know
A rational number is a fraction that has only integers for both the numerator and denominator. Division by 0 is undefined, so the denominator cannot be 0.
A rational expression is also a fraction, but it has a polynomial for both the numerator and denominator.
A non-permissible value is a value for the variable that makes the denominator of a rational expression equal to 0.
Non-permissible values are stated as restrictions on the variable to ensure the expression is defined. They must be found before you start to simplify a rational expression. Restrictions are stated as part of the answer, whether you are asked for them or not.

19. You’re ready! Try the homework from this section.
Need to Know
A rational expression is in simplified form when the numerator and the denominator have no common factors other than 1.
Steps for simplifying a rational expression:
factor the numerator and/or denominator
Find the non-permissible values
Cancel factors common to numerator and denominator and state simplified expression with NPV’s.
Equivalent rational expressions are those that simplify to the same expression.
You’re ready! Try the homework from this section.