9.5 Adding and Subtracting Rational Expressions
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Transcript of 9.5 Adding and Subtracting Rational Expressions
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9.5 Adding and Subtracting Rational Expressions
4/23/2014
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Same Denominator:
When adding or subtracting fractions with same denominators, add/subtract the numerators and
keep the same denominator
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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
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Different Denominators:
If the denominators are different, find the least common denominator (LCD).
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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:
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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)
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
c. 3 ,x 2–
xx 2 4–
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
x 1+x 1–,x
2x 1+d.
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Basic Example of Adding Fractions with different denominators.
31
43
4: 4, 8, 12, 16..3: 3, 6, 9, 12…LCD: 12
33 ∙ ∙ 4
4
+¿12
= ❑12
¿9 4 13
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Example 3 Add with Unlike Denominators
Perform the indicated operation.
a. 2x3 + x 2
4
Find LCD: 2 𝑥2 ∙ 𝑥𝑥 ∙ 22
3 𝑥+8
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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
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Checkpoint Add and Subtract with Unlike Denominators
Perform the indicated operation.
2x37.
x 2
2+
ANSWER
2x 2+3x 4
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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.
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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:
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9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39
Homework:
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Checkpoint Add and Subtract with Like Denominators
Perform the indicated operation.
1. 3x2
3x4
+ ANSWERx2
22. x 2–
xx 2–
– ANSWER –1
x2x
4+3.
x 4+– x 3–
ANSWERx 4+x 3+
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Subtracting PolynomialsSwitch the signs of everything that’s being subtracted and combine like terms.Ex:
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Example 1 Subtract with Like Denominators
Subtract.
xx
3+ x2
3+–
Subtract numerators. x
x3+ x
23+
– =x 3+x 2–
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Checkpoint Extra Examples
21. x 2–
xx 2–
–
ANSWER
–1
x2x
4+2.
x 4+– x 3–
x 4+x 3+
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Example 2 Subtract with Unlike Denominators
Subtract .2x 2 ––1
x 2
– x – 2 4
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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.
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Example 2 Subtract with Unlike Denominators
Subtract .2x 2 ––1
x 2
– x – 2 4
SOLUTIONThe 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
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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
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9.5 p.497 # 11, 14, 29, 31, 32, 38,40, 41
Homework: