9.5 Adding and Subtracting Rational Expressions 4/23/2014
Dec 25, 2015
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.
2x
3
2x
5+
Add numerators and simplify. 2x
3
2x
5+ =
3 5+
2x=
2x
8 ¿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.1
2x 4–1
3x 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–x
x 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.
3
1
4
3 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
∙22
3 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
+3
𝑥+5
Checkpoint Add and Subtract with Unlike Denominators
Perform the indicated operation.
2x3
7.x 2
2+
ANSWER
2x 2
+3x 4
Checkpoint Add and Subtract with Unlike Denominators
8.3
x 1–x
x 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:
Checkpoint Add and Subtract with Like Denominators
Perform the indicated operation.
1. 3x
2
3x
4+ ANSWER
x2
22.
x 2–
x
x 2–– ANSWER –1
x
2x
4+3.
x 4+–
x 3–ANSWER
x 4+
x 3+
Subtracting PolynomialsSwitch the signs of everything that’s being subtracted and combine like terms.
Ex:
Example 1 Subtract with Like Denominators
Subtract.
x
x
3+ x
2
3+–
Subtract numerators. x
x
3+ x
2
3+– =
x 3+
x 2–
Checkpoint Extra Examples
21.
x 2–
x
x 2––
ANSWER
–1
x
2x
4+2.
x 4+–
x 3–
x 4+
x 3+
Example 2 Subtract with Unlike Denominators
Subtract .2x 2 –
–1
x 2
– x – 2 4
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.
Example 2 Subtract with Unlike Denominators
Subtract .2x 2 –
–1
x 2
– x – 2 4
SOLUTION
The factors of are and and the factors of are and . So, the LCD is .
x 2
– x – 2 ( )x – 2 ( )x + 1x 2 – 4 ( )x – 2 ( )x + 2
( )x + 2( )x – 2 ( )x + 1
Example 2 Subtract with Unlike Denominators
Rewrite using LCD.
= – 2( )x + 2( )x – 2 ( )x + 1 ( )x – 2( )x + 2 ( )x + 2
( )x + 1
=Subtract numerators.
x + 2 – 2( )x + 1( )x – 2 ( )x + 1 ( )x + 2
= Distributive property
x + 2 – 2x( )x – 2 ( )x + 1 ( )x + 2
– 2
Combine like terms.=
x( )x – 2 ( )x + 1 ( )x + 2
–
( )x + 1
2x 2 –
–1
x 2
– x – 2 4
=1 2
– Factor denominators.( )x – 2 ( )x + 1 ( )x – 2 ( )x + 2
9.5 p.497 # 11, 14, 29, 31, 32, 38,40, 41
Homework: