1. Simplify a rational expression
EXAMPLE 1
Simplify a rational expression
x2 – 2x – 15
x2 – 9
Simplify :
SOLUTION
x2 – 2x – 15
x2 – 9
(x +3)(x –5)
(x +3)(x –3) =
Factor numerator and denominator.
(x +3)(x –5)
(x +3)(x –3) =
Divide out common factor.
x – 5
x – 3 =
Simplified form
ANSWER
x – 5
x – 3

2. Standardized Test Practice
EXAMPLE 3
Standardized Test Practice
SOLUTION
8x 3y
2x y2
7x4y3 4y
56x7y4
8xy3 =
Multiply numerators and denominators.
x x6 y3 y
8 x y3 =
Factor and divide out common factors.
7x6y =
Simplified form
The correct answer is B.
ANSWER

3. Multiply rational expressions
EXAMPLE 4
Multiply rational expressions
x2 + x – 20 3x
3x –3x2
x2 + 4x – 5
Multiply:
SOLUTION
x2 + x – 20 3x
3x –3x2
x2 + 4x – 5
3x(1– x)
(x –1)(x +5) =
(x + 5)(x – 4) 3x
Factor numerators and denominators.
3x(1– x)(x + 5)(x – 4) =
(x –1)(x + 5)(3x)
Multiply numerators and denominators.
3x(–1)(x – 1)(x + 5)(x – 4) =
(x – 1)(x + 5)(3x)
Rewrite 1– x as (– 1)(x – 1).
3x(–1)(x – 1)(x + 5)(x – 4) =
(x – 1)(x + 5)(3x)
Divide out common factors.

4. Multiply rational expressions
EXAMPLE 4
Multiply rational expressions
= (–1)(x – 4)
Simplify.
= –x + 4
Multiply.
ANSWER
–x + 4

5. Multiply a rational expression by a polynomial
EXAMPLE 5
Multiply a rational expression by a polynomial
Multiply:
x + 2
x3 – 27
(x2 + 3x + 9)
SOLUTION
x + 2
x3 – 27
(x2 + 3x + 9) =
x + 2
x3 – 27
x2 + 3x + 9 1
Write polynomial as a rational expression.
(x + 2)(x2 + 3x + 9)
(x – 3)(x2 + 3x + 9) =
Factor denominator.
(x + 2)(x2 + 3x + 9)
(x – 3)(x2 + 3x + 9) =
Divide out common factors. =
x + 2
x – 3
Simplified form
ANSWER
x + 2
x – 3

6. Divide rational expressions
EXAMPLE 6
Divide rational expressions
Divide : 7x
2x – 10
x2 – 6x
x2 – 11x + 30
SOLUTION 7x
2x – 10
x2 – 6x
x2 – 11x + 30 7x
2x – 10
x2 – 6x
x2 – 11x + 30 =
Multiply by reciprocal. 7x
2(x – 5) =
(x – 5)(x – 6)
x(x – 6)
Factor. =
7x(x – 5)(x – 6)
2(x – 5)(x)(x – 6)
Divide out common factors. 7 2 =
Simplified form
ANSWER 7 2

7. Divide a rational expression by a polynomial
EXAMPLE 7
Divide a rational expression by a polynomial
Divide :
6x2 + x – 15
4x2
(3x2 + 5x)
SOLUTION
6x2 + x – 15
4x2
(3x2 + 5x)
6x2 + x – 15
4x2
3x2 + 5x = 1
Multiply by reciprocal.
(3x + 5)(2x – 3)
4x2 =
x(3x + 5) 1
Factor.
(3x + 5)(2x – 3) =
4x2(x)(3x + 5)
Divide out common factors.
2x – 3
4x3 =
Simplified form
ANSWER
2x – 3
4x3

8. Add or subtract with like denominators
EXAMPLE 1
Add or subtract with like denominators
Perform the indicated operation. 7 4x + 3 a. 2x
x + 6 – 5 b.
SOLUTION 7 4x + 3 a. =
7 + 3 4x 10 4x = 5 2x =
Add numerators and simplify result. 2x
x + 6 5 – b.
x + 6
2x – 5 =
Subtract numerators.

9. EXAMPLE 2
Find a least common multiple (LCM)
Find the least common multiple of 4x2 –16 and 6x2 –24x + 24.
SOLUTION
STEP 1
Factor each polynomial. Write numerical factors as products of primes.
4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2)
6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2

10. EXAMPLE 2
Find a least common multiple (LCM)
STEP 2
Form the LCM by writing each factor to the highest power it occurs in either polynomial.
LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2

11. Add with unlike denominators
EXAMPLE 3
Add with unlike denominators
Add:
9x2 7 + x
3x2 + 3x
SOLUTION
To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that 9x2 = 32x2 and 3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x 1 1). 7
9x2 x
3x2 + 3x = +
3x(x + 1)
Factor second denominator. 7
9x2
x + 1 +
3x(x + 1) x 3x
LCD is 9x2(x + 1).

12. Add with unlike denominators
EXAMPLE 3
Add with unlike denominators
7x + 7
9x2(x + 1)
3x2 + =
Multiply.
3x2 + 7x + 7
9x2(x + 1) =
Add numerators.

13. Subtract with unlike denominators
EXAMPLE 4
Subtract with unlike denominators
Subtract:
x + 2
2x – 2
–2x –1
x2 – 4x + 3 –
SOLUTION
x + 2
2x – 2
–2x –1
x2 – 4x + 3 –
x + 2
2(x – 1)
– 2x – 1
(x – 1)(x – 3) – =
Factor denominators.
x + 2
2(x – 1) =
x – 3 –
– 2x – 1
(x – 1)(x – 3) 2
LCD is 2(x  1)(x  3).
x2 – x – 6
2(x – 1)(x – 3)
– 4x – 2 – =
Multiply.

14. Subtract with unlike denominators
EXAMPLE 4
Subtract with unlike denominators
x2 – x – 6 – (– 4x – 2)
2(x – 1)(x – 3) =
Subtract numerators.
x2 + 3x – 4
2(x – 1)(x – 3) =
Simplify numerator. =
(x –1)(x + 4)
2(x – 1)(x – 3)
Factor numerator. Divide out common factor.
x + 4
2(x –3) =
Simplify.

15. Simplify a complex fraction (Method 2)
EXAMPLE 6
Simplify a complex fraction (Method 2) 5
x + 4 1 + 2 x
Simplify:
SOLUTION
The LCD of all the fractions in the numerator and denominator is x(x + 4). 5
x + 4 1 + 2 x 5
x + 4 1 + 2 x =
x(x+4)
Multiply numerator and denominator by the LCD.
x + 2(x + 4) 5x =
Simplify. 5x
3x + 8 =
Simplify.

16. Solve a rational equation by cross multiplying
EXAMPLE 1
Solve a rational equation by cross multiplying
Solve: 3
x + 1 = 9
4x + 1 3
x + 1 = 9
4x + 1
Write original equation.
3(4x + 5) = 9(x + 1)
Cross multiply.
12x + 15 = 9x + 9
Distributive property
3x + 15 = 9
Subtract 9x from each side.
3x = – 6
Subtract 15 from each side.
x = – 2
Divide each side by 3.
The solution is –2. Check this in the original equation.
ANSWER