Lesson Vocabulary

• complex fraction

Problem 1 Multiplying Rational Expressions

What is the product? State any excluded values.

1. 6 a 2 ⋅ − 2 a 3 fraction 6 , over eh squared end fraction , dot . fraction negative 2 , over eh cubed end fraction

   6 a 2 ⋅ − 2 a 3 = 6 ( − 2 ) a 2 ( a 3 ) Multiply numerators and multiply denominators .   − 12 a 5 Simplify . table with 2 rows and 3 columns , row1 column 1 , fraction 6 , over eh squared end fraction , dot . fraction negative 2 , over eh cubed end fraction , column 2 equals . fraction 6 open negative 2 close , over eh squared , open , eh cubed , close end fraction , column 3 cap multiplynumeratorsandmultiplydenominators . . , row2 column 1 , , column 2 fraction negative 12 , over eh to the fifth end fraction , column 3 cap simplify , . , end table

   The product is − 12 a 5 , fraction negative 12 , over eh to the fifth end fraction . comma  where a ≠ 0 . eh not equal to 0 .

   Think

   Are products of rational expressions defined for all real numbers?

   No. The products may have excluded values.

   In part (A), the excluded value is 0. In part (B), the excluded values are 0 and − 3 negative 3 .

2. x − 7 x ⋅ x − 5 x + 3 fraction x minus 7 , over x end fraction . dot . fraction x minus 5 , over x plus 3 end fraction

   x − 7 x ⋅ x − 5 x + 3 = ( x − 7 ) ( x − 5 ) x ( x + 3 )   Multiply numerators and multiply denominators . Leave the product in factored form . fraction x minus 7 , over x end fraction . dot . fraction x minus 5 , over x plus 3 end fraction . equals . fraction open x minus 7 close open x minus 5 close , over x open x plus 3 close end fraction . . table with 2 rows and 1 column , row1 column 1 , cap multiplynumeratorsandmultiplydenominators . . , row2 column 1 , cap leavetheproductinfactoredform . . , end table

The product is ( x − 7 ) ( x − 5 ) x ( x + 3 ) , fraction open x minus 7 close open x minus 5 close , over x open x plus 3 close end fraction . comma  where x ≠ 0 x not equal to 0  and x ≠ − 3 . x not equal to negative 3 .

✓ Got It?

1. What is the product? State any excluded values.
   1. 5 y ⋅ 3 y 3 5 over y , dot , fraction 3 , over y cubed end fraction
   2. x x − 2 ⋅ x + 1 x − 3 fraction x , over x minus 2 end fraction . dot . fraction x plus 1 , over x minus 3 end fraction