objective swbat simplify rational expressions, add, subtract, multiply, and divide rational...
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ObjectiveObjective
SWBAT simplify rational SWBAT simplify rational expressions, add, subtract, multiply, expressions, add, subtract, multiply, and divide rational expressions and and divide rational expressions and solve rational equationssolve rational equations
Section 12.3 “Dividing Section 12.3 “Dividing Polynomials”Polynomials”
Divide 5xDivide 5x² -10x +4 by -5x.² -10x +4 by -5x.
““Think”Think”
x
xx
5
4105 2
x
x
5
5 2
x
x
5
10
x5
4
xx
5
42
+ +
Simplify
x
x
5
5 2
x
x
5
10
x5
4
+ +2
(6x(6x³ + 3x² - 12x) ÷ 3x.³ + 3x² - 12x) ÷ 3x.
““Think”Think”
x
xxx
3
1236 23 x
x
3
6 3
x
x
3
3 2
x
x
3
12
42 2 xx
+ +
Simplify
x
x
3
6 3
x
x
3
3 2
x
x
3
12+ +
x
Solve
2x² -4
Section 12.4 Section 12.4 “Simplify Rational Expressions”“Simplify Rational Expressions”
A A RATIONAL EXPRESSIONRATIONAL EXPRESSION is is an expression that can be written as a an expression that can be written as a ratio (fraction) of two polynomials ratio (fraction) of two polynomials where the denominator is not 0.where the denominator is not 0.
3
2
x 58
272
ww
w
Simplify Rational ExpressionsSimplify Rational Expressions
To simplify a rational expression, you can To simplify a rational expression, you can factor the numerator and denominator factor the numerator and denominator and then divide out any common factors. and then divide out any common factors.
A rational expression is in A rational expression is in SIMPLEST SIMPLEST FORMFORM if the numerator and denominator if the numerator and denominator have no factors in common other than 1. have no factors in common other than 1.
34
12
x
xxxx
x
4
342
3
x
Simplify the rational expression, if possible. Simplify the rational expression, if possible.
86
1032
2
xx
xx)2)(4( xx
FactorFactor )2)(5( xx
4
5
x
x
103
452
2
zz
zz)2)(5( zz
FactorFactor
)1)(5( zz
2
1
z
z
Recognize Recognize oppositesopposites
)2)(5(
)1)(5(
zz
zz
Multiply by -1Multiply by -1Rewrite (z-5) as -(z+5)Rewrite (z-5) as -(z+5)
Section 12.5 Section 12.5 “Multiply and Divide Rational “Multiply and Divide Rational
Expressions”Expressions” Multiplying and dividing rational Multiplying and dividing rational
expressions is similar to multiplying and expressions is similar to multiplying and dividing fractions.dividing fractions.
Be sure to simplify your answer. Look to Be sure to simplify your answer. Look to cancel like terms when multiplying or cancel like terms when multiplying or dividing. dividing.
bd
ac
d
c
b
a
bc
ad
c
d
b
a
d
c
b
a
Multiply by reciprocal,Multiply by reciprocal,then look to cancel terms.then look to cancel terms.
5
33
8
15
5
2
y
y
y
y
r
r
5
12
3
16 2
Multiply by reciprocalMultiply by reciprocal
Find the product or quotient.Find the product or quotient.
EXAMPLE 1EXAMPLE 1
6
6
40
30
y
y 12
5
3
16 2 rr
4
3 36
80 3r
9
20 3r
xx
xx
xx
xx
2
2
2
2 34
36244
33
Find the product.Find the product.
EXAMPLE 2EXAMPLE 2
Multiply numerator and denominator.Multiply numerator and denominator.
))(36244(
)34)(33(22
22
xxxx
xxxx
)1()3)(3(4
)1)(3)(1(3
xxxx
xxxx Factor and look for common Factor and look for common factors to cancel.factors to cancel.
)3(4
)1(3
x
x
87
1
32
7722
2
xx
x
xx
xx
Find the quotient. Try it out.Find the quotient. Try it out.
EXAMPLE 2EXAMPLE 2
Multiply numerator Multiply numerator and denominator.and denominator.
)1)(32(
)87)(77(2
22
xxx
xxxx
)1)(1)(3(
)1)(8)(1(7
xxx
xxxxFactor and look for common Factor and look for common factors to cancel.factors to cancel.
)3(
)8(7
x
xx
1
87
32
77 2
2
2
x
xx
xx
xx Multiply by reciprocalMultiply by reciprocal
)3(65
52
xxx
x
Find the product. Find the product.
EXAMPLE 3EXAMPLE 3
Multiply numerator and denominator.Multiply numerator and denominator.
1
3
65
52
x
xx
x
65
)3(52
xx
xx Factor and look for common Factor and look for common factors to cancel.factors to cancel.
)2)(3(
)3(5
xx
xx
2
5
xx
Section 12.6 Section 12.6 “Add and Subtract Rational Expressions”“Add and Subtract Rational Expressions”
Adding and subtracting rational Adding and subtracting rational expressions is similar to adding and expressions is similar to adding and subtracting fractions.subtracting fractions.
Be sure to simplify your answer. Be sure to simplify your answer.
c
ba
c
b
c
a
c
ba
c
b
c
a
Denominator must be COMMON!!!Denominator must be COMMON!!!
82
1
103
422
xx
x
xx
x
Find the sum or difference.Find the sum or difference. EXAMPLE 5EXAMPLE 5
)2)(5(
4
xx
x
LCD = (x+5)(x – 2)(x + 4)LCD = (x+5)(x – 2)(x + 4)
)4(
)4(
x
x
)4)(2(
1
xx
x
)5(
)5(
x
x
)4)(2)(5(
)5)(1()4)(4(
xxx
xxxx
)4)(2)(5(
)54(168 22
xxx
xxxx
)4)(2(
1
)2)(5(
4
xx
x
xx
x
)4)(2)(5(
214
xxx
x
Section 12.7 Section 12.7 “Solve Rational Equations”“Solve Rational Equations”
A A RATIONAL EQUATIONRATIONAL EQUATION is an is an equation that contains one or more equation that contains one or more rational expressions. rational expressions.
One method for solving a rational equation One method for solving a rational equation is to use the cross products property. (You is to use the cross products property. (You can use this method when both sides of the can use this method when both sides of the equation are single rational expressions). equation are single rational expressions).
24
6 x
x
32
5 y
y
yy 215 2
Cross multiplyCross multiply
Solve the equation. Check your solution.Solve the equation. Check your solution.
)2()3)(5( yy
1520 2 yy
3;5
)3)(5(0
y
yy
Check for extraneous solutions
32
5 y
y
3
3
5
53
3
23
5
32
5 y
y
3
5
3
53
5
25
5
y = 5y = 5 y = -3y = -3
Chapter 8 Chapter 8 Final ReviewFinal Review
4
3 512 yx9)7(
5)5)
7)7)
3)3)
12x1)1)
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8; xd9)9)
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1
ba2)2)
236p4)4)
3
312
c
d
6)6)
5
10
27x
y
8)8)
144
1;d
10)10)
12
6
64x
y
Chapter 9 Chapter 9 Final ReviewFinal Review
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9419 2 nn
5)5)
7)7)
3)3)
792 2 aa1)1)
)1)(7( xx9)9)
778 23 xxx2)2)
11559 23 ccc4)4)
6352 2 zz
652 23 bbb
8)8)
)11)(43( xx10)10)
6)6)2972144 gg
2
5,0a
11)11)
11,11,0 n12)12)
Chapter 10 Chapter 10 Final ReviewFinal Review
A
3,3 w
5)5)
7)7)
3)3)
6,6 x
1)1)
D9)9)
parabola
2)2)
)1)(3( pp4)4)
1.4;1.1
5.3;1
8)8)
B10)10)
6)6) 91.0;11.1
106
7
5
23
152
n
n
5337
cdc 34 32
3)3)
4)4)
7)7)
8)8)
2)2)
241)1)
36
78
5)5)
6)6)
Chapter 11 Review worksheetChapter 11 Review worksheet
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solutionno10)10)
11x11)11)
7x12)12)
13413)13)
53d14)14)
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