The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 6.5: Simplifying Algebraic Fractions

Some students incorrectly apply the cancellation rule when simplifying an algebraic fraction.

Fully understanding the process eliminates this common error.

1. Explain that some algebraic fractions, fractions that have a variable in the numerator and/or
   denominator, may be simplified using the same process that is used for simplifying fractions.


2. Present this example to your students:^10
   15

2

3

2 · 5

3 · 5

==Note that the common factor, 5,

may be cancelled because

5

5

=1. Explain that this same process applies to simplifying the

quotients of polynomials by factoring and cancelling the common factors, provided that the

denominator does not equal 0. If the denominator equals 0, the fraction is undefined.

3. Review the information and examples on the worksheet with your students. Explain that
   only the factors of a polynomial that have the same terms may be cancelled. Note the first
   example:

8 + 4 x

6 x+ 12

=

4(2+x)

6(x+2)

.Inthisexample,

2 +x

x+ 2

can be cancelled because addition is

commutative and this fraction is equal to 1. Emphasize that in the second example students

cannot cancel thex^2 term becausex^2 is not a factor of the polynomial. Students must factor

the trinomial.

EXTRA HELP:

The quotient of polynomials is in simplest form if the only common factor is 1 or−1.

ANSWER KEY:

(1)

x− 4

6 x− 2

(2)

1

2

(3)

5

6

(4)

2 x+ 1

6 x+ 4

(5)x+y (6)

x+ 3

x+ 4

(7)

3 x− 4

1 +x

(8)x− 5

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(Challenge)No. It can be simplified further as

2 x− 8

x+ 5

.

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232 THE ALGEBRA TEACHER’S GUIDE