2.4. Rational Expressions http://www.ck12.org
((x(+(^3 )((x(−(^3 ()
(x^2 + 9 )((x(+( 3 )((x(−( 3 ()−4 x
(x^2 + 9 )·2
(x− 3 )=(x (^2) +^19 )−(x (^2) + 98 )(xx− 3 )
=(x (^2) +(x 9 −)(^3 x)− 3 )−(x (^2) + 98 )(xx− 3 )
=(x 2 (+−^79 x)(−x^3 −) 3 )
The numerator cannot factor at this point, so in this example there is not a factor that cancels at the end. Remember
that the sum of perfect squares does not factor.
Example C
Combine the following rational expressions.
1
x^2 + 5 x+ 6 −
1
x^2 − 4 +(x− 7 )(x+ 5 )+ 5
(x+ 2 )(x− 2 )(x+ 3 )(x− 4 )Solution: First factor everything and decide on a common denominator. While the numerators do not really need
to be factored, it is sometimes helpful in simplifying individual expressions before combining them. Note that the
numerator of the expression on the right hand seems factored but it really is not. Since the 5 is not connected to
the rest of the numerator through multiplication, that part of the expression needs to be multiplied out and like terms
need to be combined.
=(x+ 2 )(^1 x+ 3 )−(x+ 2 )(^1 x− 2 )+ x(^2) − 2 x− 35 + 5
(x+ 2 )(x− 2 )(x+ 3 )(x− 4 )
=(x+ 2 )(^1 x+ 3 )−(x+ 2 )(^1 x− 2 )+ x
(^2) − 2 x− 30
(x+ 2 )(x− 2 )(x+ 3 )(x− 4 )
Note that the right expression has 4 factors in the denominator while each of the left expressions have two that match
and two that are missing from those four factors. This tells you what you need to multiply each expression by in
order to have the denominators match up.
=(x+ 2 )((xx−−^22 )()(xx+−^43 )()x− 4 )−(x+ 2 )((xx+−^32 )()(xx+−^43 ))(x− 4 )+ x
(^2) − 2 x− 30
(x+ 2 )(x− 2 )(x+ 3 )(x− 4 )
Now since the rational expressions have a common denominator, the numerators may be multiplied out and com-
bined. Sometimes instead of rewriting an expression repeatedly in mathematics you can use an abbreviation. In
this case, you can replace the denominator with the letterDand then replace it at the end.
=(x−^2 )(x−^4 )−(x+^3 )(x−^4 )+x
(^2) − 2 x− 30
D
=x
(^2) − 6 x+ 8 −[x (^2) −x− 12 ]+x (^2) − 2 x− 30
D
Notice how it is extremely important to use brackets to indicate that the subtraction applies to all the terms of the
middle expression not justx^2. This is one of the most common mistakes.