9.5 Adding and Subtracting Rational Expressions
4/23/2014
Same Denominator:
When adding or subtracting fractions with same denominators, add/subtract the numerators and
keep the same denominator
Example 1 Add and Subtract with Like Denominators
Perform the indicated operation.
2x3
2x5
+
Add numerators and simplify. 2x
32x5
+ =3 5+
2x=
2x8 ¿
4 ∙22 ∙𝑥 ¿
𝟒𝒙
Different Denominators:
If the denominators are different, find the least common denominator (LCD).
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
a. 6x1 ,
8x 3
1
List the multiples of the number:6: 6, 12, 18, 24, 30, 36….8: 8, 16, 24, 32…Take the Least Common Multiple (LCM): 24As for the variable, take the one with highest exponent: LCD:
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
b. 12x 4–
13x 9+
,
2x - 4 = 2(x – 2)3x + 9 = 3(x + 3)2: 2, 4, 6, 8…3: 3, 6, 9, 12…LCD: 6(x-2)(x+3) (Take the LCM and multiply by the rest of their factors)
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
c. 3 ,x 2–
xx 2 4–
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
x 1+x 1–,x
2x 1+d.
Basic Example of Adding Fractions with different denominators.
31
43
4: 4, 8, 12, 16..3: 3, 6, 9, 12…LCD: 12
33 ∙ ∙ 44
+¿12
= ❑12
¿9 4 13
Example 3 Add with Unlike Denominators
Perform the indicated operation.
a. 2x3 + x 2
4
Find LCD: 2 𝑥2 ∙ 𝑥𝑥 ∙ 223 𝑥+8
Example 3 Add with Unlike Denominators
Find LCD: (𝑥−2)(𝑥+5) ∙ (𝑥+5)
(𝑥+5) ∙ (𝑥−2)(𝑥−2)
2 𝑥2+10 𝑥+3 𝑥−6
¿2𝑥2+13 𝑥−6(𝑥−2)(𝑥+5)
𝑏 . 2𝑥𝑥−2+
3𝑥+5
Checkpoint Add and Subtract with Unlike Denominators
Perform the indicated operation.
2x37.
x 2
2+
ANSWER
2x 2+3x 4
Checkpoint Add and Subtract with Unlike Denominators
8. 3x 1–
xx 1++
ANSWER
x 2 + 2x + 3( )x – 1 ( )x + 1
Perform the indicated operation.
Example 4 Add with Unlike Denominators
Find LCD: (𝑥+6)(𝑥−6)
∙ (𝑥−6)(𝑥−6)
2 𝑥+3 𝑥−18
¿5 𝑥−18
(𝑥−6)(𝑥+6)
Add:
9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39
Homework:
9.5 cont’d….Subtracting Rational Expressions
4/25/2014
Example 1 Subtract with Like Denominators
Subtract.
xx
3+ x2
3+–
Subtract numerators. xx
3+ x2
3+– =
x 3+x 2–
Example 2 Subtract with Like Denominators
Perform the indicated operation.
2x 2–
xx 2–
– = = =
Subtracting PolynomialsSwitch the signs of everything that’s being subtracted and combine like terms.Ex:
Example 3 Subtract with Like Denominators
Perform the indicated operation.
= = =x
2x4+ x 4+
– x 3–
Example 4 Subtract with Unlike Denominators
Subtract
LCD:
= 𝑥2 (𝑥+3 )−3 𝑥 (𝑥−2)
∙ 𝑥+3𝑥+3
=
=
∙ 𝑥−2𝑥−2
𝑥2
𝑥−2− 3 𝑥𝑥+3
=
Example 5 Subtract with Unlike Denominators
Subtract 2x 2 ––1
x 2
– x – 2 4
-2
-1
1 -2
=
LCD:
= 1 (𝑥+2 )−2(𝑥+1)
∙ 𝑥+2𝑥+2
=
=
∙ 𝑥+1𝑥+1
Checkpoint Add and Subtract with Unlike Denominators
1 1–x 2
– 1 x + 1
ANSWER
– x – 2( )x – 1( )x + 1
Perform the indicated operation.
9.5 p.497 # 11, 14, 17-23odd, 29, 31, 32, 38,40, 41
Homework: