Adding and Subtracting Rational Expressions

Key Concepts

• Add rational expressions with like denominators.
• Identify the LCM of polynomials.
• Add rational expressions with unlike denominators.
• Subtract rational expressions.
• Simplify a compound fraction.

Add Rational Expressions with Like Denominators 

 What is the sum? 

1. x/x+4 + x/x+5  

= x+5/x+4———————————– When denominators are the same, add the numerators.  

So, 

x/x+4 + 5/x+4 = x+5/x+4

2. 2x+1/x²+3x + 3x −8x/x(x+3)  

= (2x+1)+(3x −8)/x²+3x———————- Add the numerators.  

= (2x+ 3x)+(1 −8)/x²+3x——————— Use the Commutative and Associative Properties.  

= 5x −7/x²+3x——————————– Combine like terms.  

So, 

2x+1/x²+3x = 3x −8/x(x+3) = 5x −7/x²+3x

Identify the LCM of Polynomials  

How can you find the least common multiple of polynomials?  

1. (x + 2)², x² + 5x + 6 

 Factor each polynomial.  

(x + 2)² = (x + 2) (x + 2) 

x² + 5x + 6 = (x + 2) (x + 3) 

The LCM is the product of the factors. Duplicate factors are raised to the greatest power represented.  

LCM: (x + 2) (x + 2) (x + 3) or (x + 2)²(x + 3) 

2. x³ – 9x, x² – 2x – 15, x² – 5x  

Factor each polynomial.  

x³ – 9x = x (x² – 9) = x (x + 3) (x – 3)  

x² – 2x – 15 = (x + 3) (x – 5) 

x² – 5x = x (x – 5) 

LCM: x (x + 3) (x – 3) (x – 5).  

Add Rational Expressions with Unlike Denominators  

What is the sum of x + 3/x² − 1  and 2/x²−3x+2 

Follow a similar procedure to the one you use to add numerical fractions with unlike denominators.   

The sum of

x + 3/x² − 1  and 2/x²−3x + 2   is

x + 4/(x + 1)(x − 2) for x ≠ –1, 1, and 2.   

Subtract Rational Expressions 

What is the difference between x +1/x² – 6x – 16  and x+1/x² + 6x + 8  ?

The difference between x +1/x² – 6x – 16 and x+1/x² + 6x + 8   is

12(x +1)/(x−8)(x+2)(x+4) for x ≠ -4, -2, and 8.   

Application: Find a Rate 

Leah drives a car to the mechanic, then she takes the commuter rail train back to her neighborhood. The average speed for the 10-mile trip is 15 miles per hour faster on the train. Find an expression for Leah’s total travel time. If she drove 30 mph, how long did this take? 

Solution: 

Total time for the trip:  

10/r +10/r+15 = 10(r+15)/r(r+15) + 10r/r(r+15)

=10r + 150 + 10r/r(r+15) = 20r +150/r(r+15)

At a driving rate of 30 mph, you can find the total time.  

20r+150/r(r+15) = 20(30)+150/30(30+15)

               = 750/1350

               = 5/9

The expression for Leah’s total travel time is 20r+150 r(r+15)

The total time is 5/9 h, or about 33 min.  

Compound Fraction 

Simplify a Compound Fraction 

A compound fraction is in the form of a fraction and has one or more fractions in the numerator and the denominator. How can you write a simpler form of a compound fraction? 

Method 1:  

Find the Least Common Multiple (LCM) of the fractions in the numerator and the denominator. Multiply the numerator and the denominator by the LCM.  

Method 2:  

Express the numerator and denominator as single fractions. Then multiply the numerator by the reciprocal of the denominator.  

Questions  

Question 1 

Find the sum.  

1. 10x − 5/2x + 3 + 8 − 4x/2x +3

2. x + 6/x² − 4 + 2/x² − 5x + 6

Solution: 

1.  

Question 2 

Simplify:  

3x − 5/x² − 2/x + 5

Solution:  

Question 3 

Simplify the compound fraction:  

Solution: 

Key Concepts Covered