Hon Alg 2: Unit 2
What is a RATIONAL EXPRESSION?
A RATIONAL EXPRESSION is an algebraic FRACTION whose numerator and denominator are polynomials.
What are EXCLUDED VALUES a rational expression? (Never divide by a ZERO)
EXCLUDED VALUES are any values for the variable that results in a denominator equal to zero.
To Find Excluded Values: Set DENOMINATOR = 0 and solve the equation .
(Hint: Factor Denominator as needed to solve)
5m  3
x2  5
1) 10a  5
3)
2)
3a
( x  2)( x  3)
m6
y 2  2 y  15
4) 2
y  y  12
How do you simplify rational expressions? (REDUCE FRACTIONS)
Step #1: FACTOR the Numerator and the Denominator
Step #2: CANCEL any common factors in both numerator and denominator or apply laws of exponents
HELPFUL Property of –1:
1  x  x  1 (1)( x  1)


 1
x 1
x 1
x 1
1)
90 x 2 y 3
12 x 4 y
2)
 7a 2 b 6
21a 5 b 3
3)
3x 2  9x
3 x
4)
6( x  3)( x  2)( x  5)
=
2( x  5)( x  2)( x  3)
5)
9( x  7 )( 3 x  2)( 2 x  5)
=
7( 2 x  5)( 7  x )
6)
5 y 3 ( 2 y  3)( y  1)

25 y( y  1)( 2 y  3)
a 2b  a
7) 3
c  c 3b
10)
5 x 3  40
x2  4
3 x 2  12 x  36
8)
x2
11)
2x 2  5x  7
x 2  6x  5
y 2  2 y  15
9)
y 2  y  12
12)
x 2  3 x  28
3 x 2  13 x  12
PRACTICE PROBLEMS: State the excluded values and then simplify, if possible
35 yz 2
28 y 2 z
2)
16 s 7 r 6
64r 6 s
6 p3  9 p2
4)
21 p( 2 p  3)
5)
15  5n
( n  3)( n  5)
a 2  2a  1
7) 2
a  2a  3
x 2  4 x  12
8) 2
x  2x  8
1)
3)
4a
3a
6)
3n  18
n 2  36
2 x 2  x  21
9)
2 x 2  15 x  28
MULTIPLYING RATIONAL EXPRESSIONS:
Step #1: FACTOR the Numerators and Denominators of each fraction
Step #2: MULTIPLY ALL Numerators and ALL Denominators to make a one fraction (Use Parentheses)
Step #3: SIMPLIFY by canceling any common factors
EXAMPLES: Simplify each of the following rational expression multiplications
11 pr 5 12 p 3
y2 x5z4
36
5ab 3 16c 2

3)

1)
2) 3 8 
3
2
2
7
q
33q 4 r
42
8c
15a b
x y z
4)
(m  4)
4m 2

(m  4)( m  5)
3m
5)
( x  1)( x  1) ( x  3)( x  4)

( x  2)( x  3) ( x  3)( x  1)
6)
35 y 3 ( 2 y  1)( y  4) ( y  6)( y  4)

( y  6)( y  6)
70 y( 2 y  1)
x5
63 x 2

7)
35 x x 2  2 x  15
a 2  7a  10 3a  3

8)
a1
a2
b 2  5b  6
7
 2
9) 2
3b  8b  4 b  4b  3
PRACTICE PROBLEMS: Simplify the multiplications
24 x 5
( x 2  9)
a5
(a  2)( a  3)
12 xy 2 27m 3 p

3)

2)

1)
( x 3  27) 10 x 3
(a  2)( a  5)
5
45mp 2 20 x 3 y
x 2  10 x  21 5 x  10
 2
4)
6x  9
x  x6
4 y 2  11 y  6
y 2  16

5)
2 y2  8
4 y 2  11 y  6
b 2  10b  24 b 2  4b  5

6)
b 2  5b  6 b 2  7b  12
3b 2  8b  4
 5b  5
 2
7) 2
b  7b  8 5b  11b  6
Hon Alg 2: Unit 2
DIVIDING RATIONAL EXPRESSIONS:
Step #1: CHANGE Division into Multiplication by the RECIPROCAL of the divisor

Reciprocal of fraction
A
B
=
(FLIP)
A
B
Step #2: FACTOR the Numerators and Denominators of each fraction
Step #3: MULTIPLY ALL Numerators and ALL Denominators to make a one fraction (use Parentheses)
Step #4: SIMPLIFY by canceling any common factors
EXAMPLES: Simplify each of the following rational expression divisions
2
1)
5x
10 x 3

7
21
4)
n  1 2n  2

n 3 n4
7)
a 2  3a  2 a  2

4
a 1
2)
5)
8 y2
 24 y
9
5a  10
a2
a5
x 3  8 x 2  15 x
x ( x  3)
 2
8)
11( x  4)
x  x  20
3)
12 x 3 z 6
4x5

35 y 6
21 y 4 z 3
6)
( x  2)( x  3) ( x  3)

7( x  1)
( x  1)
2b 2  7b  6 b  2

9)
3b 2  11b  8 b  1
PRACTICE PROBLEMS: Simplify the quotient
3
2
1)
4x
8x
 2
4
y
y
5)
25a 9
15a 3 b 5

18b 4 c 2
4c 8
2)
12( y  3)( x  1) 6( y  3)( x  1)

5( x  5)( y  3) 25( y  2)( x  5)
6)
( x  1)( x  1) ( x  3)( x  1)

( x  2)( x  3) ( x  3)( x  4)
x2  2x  1 x  1

3)
2
x 1
x 2  8 x  16 2 x  8

7) 2
x  6x  9 3x  9
x 2  7 x  10 x 2  4
 3
4)
2
x 8
8)
3 x 2  10 x  7 5 x 2  7 x  2

x 2  8 x  16
x 2  16