## 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/x^{2}+3x + 3x −8x/x(x+3)

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

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

= 5x −7/x^{2}+3x——————————– Combine like terms.

So,

2x+1/x^{2}+3x = 3x −8/x(x+3) = 5x −7/x^{2}+3x

### Identify the LCM of Polynomials

**How can you find the least common multiple of polynomials? **

1. (x + 2)^{2}, x^{2 }+ 5x + 6

^{ }Factor each polynomial.

(x + 2)^{2} = **(x + 2) (x + 2)**

x^{2} + 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. x^{3} – 9x, x^{2} – 2x – 15, x^{2} – 5x

**Factor each polynomial. **

x^{3} – 9x = x (x^{2} – 9) = x** (x + 3) (x – 3)**

x^{2} – 2x – 15 = **(x + 3) (x ****–**** 5)**

x^{2} – 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/x^{2} − 1 and 2/x^{2}−3x+2

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

The sum of

x + 3/x^{2} − 1 and 2/x^{2}−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/x^{2} – 6x – 16 **and** x+1/x^{2} + 6x + 8 **?**

The difference between x +1/x^{2} – 6x – 16 and x+1/x^{2 }+ 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. **

- 10x − 5/2x + 3 + 8 − 4x/2x +3

- x + 6/x
^{2}− 4 + 2/x^{2}− 5x + 6

**Solution:**

**Question 2**

**Simplify: **

3x − 5/x^{2} − 2/x + 5

**Solution: **

**Question 3**

**Simplify the compound fraction: **

**Solution: **

### Key Concepts Covered

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