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Warm-up• Given these solutions below: write the
equation of the polynomial:
• 1. {-1, 2, ½)
3 22 3 3 2x x x
Rational Equations
Section 2-6
Objectives
• I can simplify rational expressions
• I can find Domain Restrictions
• I can solve rational equations with one variable
Simplifying Rational Expressions
• Try and reduce numerator over denominator
• You will factor all numerators and denominators, then
• Reduce or cancel like terms
Domain of Rational Functions
• The domain of any rational function is all real numbers except where the following happens:– No x-value that makes denominator zero– No x-value that would be a discontinuity (hole)
EXAMPLE 1 Simplify a rational expression
x2 – 2x – 15x2 – 9
Simplify :
x2 – 2x – 15x2 – 9
(x +3)(x –5)(x +3)(x –3)= Factor numerator and denominator.
(x +3)(x –5)(x +3)(x –3)= Divide out common factor.
Simplified form
SOLUTION
x – 5x – 3=
ANSWER x – 5x – 3 3,3:Re xstrictionsDomain
GUIDED PRACTICE for Examples 1 and 2
x2 – 2x – 3x2 – x – 6
5.
(x – 3)(x + 1)(x – 3)(x + 2)
x2 – 2x – 3x2 – x – 6 = Factor numerator and denominator.
Divide out common factor.
x + 1x + 2= Simplified form
=(x – 3)(x + 1)(x – 3)(x + 2)
SOLUTION
x + 1x + 2ANSWER 3,2:Re xstrictionsDomain
GUIDED PRACTICE for Examples 1 and 2
2x2 + 10x3x2 + 16x + 5
6.
2x2 + 10x3x2 + 16x + 5 (3x + 1)(x + 5)
2x(x + 5)= Factor numerator and
denominator.
Divide out common factor.
2x3x + 1= Simplified form
(3x + 1)(x + 5)2x(x + 5)=
ANSWER 2x3x + 1
SOLUTION
3
1,5:Re xstrictionsDomain
Adding & Subtracting Rational Expressions
• MUST have a COMMON DENOMINATOR
• You will factor all denominators, then find the Common Denominator
• Reduce or cancel like terms
Basic Fraction
2
1
3
2
3
2
2
1
6
6
x2
x2 4
x3
x3 3
4 + 3
6
6
7
)3)(2( xx
x
)2)(2(
2
xx
(x+2)
(x+2) x(x+2)
(x+3)
(x+3) 2(x+3)
x(x+2) - 2(x+3)
44
2
65 22
xxxx
x
)2)(3)(2( xxx
)2)(3)(2( xxx
)2)(3)(2( xxx
)2)(3)(2(
6222
xxx
xxx
)2)(3)(2(
62
xxx
x
)3(2
5
x
x
)3(4
7
x
x
2
2 2(x-5)
1
1 x - 7
2(x-5) - 1(x-7)
124
7
62
5
x
x
x
x
)3(4 x
)3(4 x
)3(4 x
)3(4
7102
x
xx
)3(4
)3(
x
x
4
1
Solving Rational Equations
• Two basic methods
• 1. Set equation equal to ZERO and then get Common Denominator
• 2. Two ratios equal means you can Cross Multiply to solve them
Cross Multiplication Method
3
36
3
24
xx
)3(24)3(36 xx
722410836 xx
18012 x
15x
Cross Multiplication Ex2
06
42
x
x
x
x
6
42
x
x
x
x
)4()6)(2( xxxx
xxxx 4128 22
124 x
3x
Set Equation to ZERO
)2(
1
6
5
)2(3
1
x
x
x
x0
)2(
1
6
5
)2(3
1
x
x
x
x
)2(3
1
x
x
6
5x
)2(
1
x
)2(6 x
)2(6 x
)2(6 x
2
2 2(x+1)
(x-2)
(x-2) 5x(x-2)
6
6 6
2(x+1) – 5x(x-2) - 6
6(x-2)
Next Slide
Problem Continued
0)2(6
6)2(5)1(2
x
xxx
0610522 2 xxx
04125 2 xx
04125 2 xx
020122 xx
0)2)(10( xx
0)5
2)(
5
10( xx
}5
2,2{
MUST CHECK ANSWERS
x = 2 does not work
}5
2{
Extraneous Solutions
• Extraneous solutions are those that do not work when you plug them back into the original equation.
• Usually they don’t work because they make the Denominator zero
Homework
• Worksheet 5-1
