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# Adding and Subtracting Rational Expressions

Sep 16, 2022

## Key Concepts

### 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/x+ 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
1. 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:

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