Rational Expressions
Common Mistakes
Rational Expressions - Simplify
How to Simplify Rational Expressions


Factor the numerator and denominator

Ex.
(x + 2) = (x + 2)
2
x − 5 x − 14 ( x + 2 )( x − 7 )
Determine what makes the problem
undefined ( x ≠ )

After the problem is factored, look at the
denominator.

Decide what makes each part of the
denominator equal to zero. That is the number
that x cannot be.
Ex. In the problem above, x cannot equal -2 or 7



Not knowing how to factor. Bottom line,
if you cannot factor,
you cannot work Rational Expressions.

Canceling incorrectly
x ( x + 3)
 Incorrect:
(x − 2)

Cancel


Common Mistakes
In the example, there is an (x + 2) in both the
numerator and denominator. Cancel them out.
Write the solution
(x + 2)
x − 5 x − 14
2
=
(x + 2) = 1
(x + 2)(x − 7 ) (x − 7 )
Complete Manual: ..\Rational Expression Review.docx
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
Canceling the x in the numerator with the x in
the (x-2) in the denominator is incorrect.

In this example, nothing cancels.
Incorrect:


(x − 3)2
( x − 3)
=1
Canceling both terms is incorrect because the
numerator has two (x-3)’s.
Correct:
(x − 3)2
( x − 3)
x−3 x−3
= ( x − 3)
= 

 x −3 





Rational Expressions – Multiply and Divide
How to Multiply Rational
Expressions

Factor the numerator and denominator


Ex.
Common Mistakes

if you cannot factor,
Determine what makes the problem undefined

After the problem is factored, look at the
denominator.

Decide what makes each part of the denominator
equal to zero. That is the number that x cannot be.

Not knowing how to factor. Bottom line,
you cannot work Rational Expressions.

Missing the − 1 that is factored out.

Ex. In the problem below, x cannot equal -2, 7, or 3

Cancel

Write the solution
Incorrect: (2 − x ) = 1
(x − 2)

(x + 2)
x2 − 6x − 7
( x − 3)
x 2 − 5 x − 14
=
=
⋅
(x + 2) ⋅ (x − 7 )(x + 1)
(x + 2)(x − 7 ) (x − 3)
(x + 1)
(x − 3)
Complete Manual: ..\Rational Expression Review.docx
To view; right click and open the hyperlink

Note:
In this case the two do not cancel
because they are not identical.
Correct:
(2 − x ) = − 1(x − 2) = −1
(x − 2) (x − 2)
To divide, flip the fraction following the
division sign (multiply by the reciprocal) and then
follow the steps for multiplication
Rational Expressions – Finding the LCD
Lowest Common Denominator
How to find the LCD


Factor the denominator
Write every part that is different, the
LCD is the product of unique factors
from the denominator.


1
c
−
c − c − 12 (2c − 8)
(c − 4 )(c + 3) 2(c − 4 )
2

The LCD is: 2(c − 4 )(c + 3)
( x − 3)
4
Ex. (x + 2)(x + 2) + (x + 2)(x − 1)



Not factoring correctly
Missing the − 1 that is factored out.

Hint: Don’t repeat, unless it appears twice
in one term.
Ex.

Common Mistakes
The LCD is: ( x + 2 )( x + 2 )( x − 1)
Complete Manual: ..\Rational Expression Review.docx
To view; right click and open the hyperlink
Incorrect:


(2 − x ) = 1
(x − 2)
In this case the two do not cancel because
they are not identical.
Correct: (2 − x ) − 1(x − 2)
= −1
=
(x − 2) (x − 2)
Rational Expressions – Add, Subtract
How to Add and Subtract Rational
Expressions

Factor each fraction, numerator and
denominator.
Common Mistakes

Not knowing how to find the LCD.

Canceling before the last step.

2
1
2
1
+
=
+
2
(x − 4) 5 x − 20 x (x − 4) 5 x(x − 4)
(


)
2
(x + 1)(x − 4)
Find the LCD. (see previous slide)

LCD : 5 x( x − 4 )

Correct:
Incorrect:
wrong
Correct:
4 ⋅ (x − 4 ) − x + 2 4 x − 16 − [x + 2] 4 x − 16 − x − 2
3 x − 18
=
=
=
(x + 1)(x − 4)
(x + 1)(x − 4)
(x + 1(x − 4)) (x + 1)(x − 4)
2 ⋅ 5x + 1
10 x + 1
=
5 x(x − 4) 5 x( x − 4 )
Complete Manual: ..\Rational Expression Review.docx
To view; right click and open the hyperlink
1
2 + 1 ⋅ ( x + 1)
3
=
=
(x − 4) (x + 1)(x − 4) (x − 4)
(x + 2) = 4 ⋅ (x − 4) − x + 2 = 4 x − 16 − x + 2 = 3x − 14
4
−
(x + 1)(x − 4) (x + 1(x − 4))
(x + 1) (x − 4)(x + 1) (x + 1)(x − 4)
Add or Subtract the numerators
Cancel if possible.
+
Making subtraction errors

2
1
2 ⋅ 5x + 1
+
=
( x − 4 ) 5 x( x − 4 ) 5 x( x − 4 )

wrong
2
1
2 + 1 ⋅ ( x + 1) 2 + x + 1 x + 3
+
=
=
=
(x + 1)(x − 4) (x − 4) (x + 1)(x − 4) (x − 4) x − 4
Multiply each fraction by the LCD to create
equivalent fractions with common denominators.


Incorrect:

Hint: Anytime you see a subtraction, put a bracket around what
follows it and you’ll be less likely to miss the signs.