8.4-8.7 Study Guide
Algebra 2B
8.4
Simplifying Rational Expressions
 Check numerators/denominators of fractions to see if they can be factored.
Problems with addition/subtraction in the numerator or denominator can often be factored.
Problems without addition/subtraction in the numerator or denominator are already “factors.”
 Simplify by “Canceling” COMMON FACTORS in the numerator and denominator.
 State Restrictions on the variables.
Set the ORIGINAL denominator equal to zero and solve. This is a restriction on x. The denominator of any
fraction is never allowed to be equal to zero.
Examples:
24 x  48
24( x  2)
24


2
Ex.
x 4
( x  2)( x  2)
x2
factor
simplify
x  2, -2
x 2  3x  10
( x  5)( x  2)
x2


x  0,5
2
Ex.
x  5x
x( x  5)
x
factor
simplify
Ex.
15 xy 3
3

x  0,y  0
3 5
25 x y
5x2 y 2
ready
simplify
Multiplying Rational Expressions
 Check numerators/denominators of fractions to see if they can be factored.
 Simplify by “Canceling” COMMON FACTORS in the numerator and denominator.
With multiplication you can cancel a factor within the same fraction or can cancel a numerator of one fraction
with the denominator of another fraction.
 State Restrictions on the variables.
Check the ORIGINAL denominators of all fractions for values of x that cause the denominator to be zero.
Dividing Rational Expressions
 Change division into multiplication by the reciprocal then follow the steps for multiplication.
 State Restrictions.
Check the ORIGINAL denominators AND the denominator of the reciprocal for values that cause the
denominator to equal zero.
Ex.
Ex.
x 2  15 x  50 20 x3
( x  5)( x  10)
20 x3
4 x2




x  0,-5,  10
5 x 2  25 x x 2  100
5 x( x  5)
( x  10)( x  10)
x  10
5 x  15 x 2  9
5( x  3) ( x  3)( x  3)
5( x  3)
15 x 3
75 x 2






x  0,2,  3
x 2  2 x 15 x 3
x( x  2)
15 x 3
x( x  2) ( x  3)( x  3)
( x  2)( x  3)
factor
restrictions: x  0,2
reciprocal
new restrictions: x  3
simplify
8.5
Adding/Subtracting Rational Expressions
 Check denominators of fractions to see if they can be factored.
Problems with addition/subtraction in the denominator can often be factored.
Problems without addition/subtraction in the denominator are already “factors.”
 Multiply numerator and denominator of each fraction by a value that will create COMMON DENOMINATORS.
 Simplify your numerators.
 Write the combined numerators over the COMMON DENOMINATOR.
 Check to see if your new numerator will factor. Simplify if possible.
 State the restrictions on the variable.
3
7
2 y 2  3  5x  7 
6 y2
35 x
6 y 2  35 x






x  0,y  0




Ex. 15 x 2 y 6 xy 3
2 y 2  15 x 2 y  5 x  6 xy 3  30 x 2 y 3 30 x 2 y 3
30 x 2 y 3
create common denom.
Ex.
 x 5

3
4
3
4
x2
3
4
 2







x  25 x  7 x  10
( x  5)( x  5) ( x  5)( x  2)
x  2  ( x  5)( x  5)  x  5  ( x  5)( x  2) 
factor , change "  " to "   "
create common denom.
2

3 x  6  4 x  20
 x  26

x  5,-5,-2
( x  5)( x  5)( x  2)
( x  5)( x  5)( x  2)
8.6
Solving Equations with Rational Expressions
 Check denominators of fractions to see if they can be factored.
Problems with addition/subtraction in the denominator can often be factored.
Problems without addition/subtraction in the denominator are already “factors.”
 Determine the Least Common Denominator (LCD) for all fractions in the problem.
 Multiply all terms in the equation by the LCD.
 Simplify to eliminate fractions.
 Solve.
 Check the restrictions on the variable for possible extraneous roots.
Ex.
Ex.
4
3
1
 2

x2 x 4 x2
4
3
1


x  2 ( x  2)( x  2) x  2
5
3

x2 x5
 5 
 3 
( x  2)( x  5) 
  ( x  2)( x  5) 

x

2


 x5
( x  5)(5)  ( x  2)(3)
5 x  25  3 x  6


3
 4 
 1 
( x  2)( x  2) 
  ( x  2)( x  2) 
  ( x  2)( x  2) 

 x2
 x2
 ( x  2)( x  2) 
( x  2)(4)  (3)  x  2
2 x  31
x  15.5
4x  8  3  x  2
4 x  11  x  2
3 x  13
13
x   x  4.3
3