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SWBAT…simplify rational expressions Wed, 4/3Agenda
1. WU (15 min)2. Lesson on rational expressions (25 min)
-6 examples
1.) Factor 7x – 72.) Factor 8x – 83.) Factor x2 – 14.) Factor x2 – 95.) Factor x2 – 3x – 106.) x3 + 3x2
HW#3: Dividing Monomials
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Simplifying Rational Expressions
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Multiplying &Dividing Rational Expressions
Remember that a rational number can be expressed as a quotient of two integers.
A rational expression can be expressed as a quotient of two polynomials.
3
1
2
13
2
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Remember, denominators can not = 0.
Examples of rational expressions
2
4 8 4 7, ,
3 3 5 9
x y
x x y y
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For what values of x is each rational expression undefined?
1
9 and ,
3
2 ,
1 2
x
x
xx
x = 0 x = -3 x = 1
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Simplify: 7x 7
x2 1
Step 1: Factor the numerator and the denominator completely looking for common factors.
7x 7 7(x 1)
x2 1 (x 1)(x 1)
Next
Ex. 1
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7x 7
x2 1
7(x 1)
(x 1)(x 1)
What is the common factor?
x 1Step 2: Divide the numerator and denominator by the common factor.
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7(x 1)
(x 1)(x 1)
7(x 1)
(x 1)(x 1)
1
1
Step 3: Multiply to get your answer.
Answer: 7
x 1
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Looking at the answer from the previous example, what value of x would make the denominator 0?
x = -1
The expression is undefined when the values make the denominator equal to 0
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1
882
x
x)1)(1(
)1(8
xx
x 1
11
8
x
54
1032
2
xx
xx
)5)(1(
)5)(2(
xx
xx
)1(
)2(
x
x1
1
Ex. 2
Ex. 3
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9
652
2
x
xx
)3)(3(
)3)(2(
xx
xx
)3(
)2(
x
x1
1
Ex. 4
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Remember how to multiply fractions:
First you multiply the numerators then multiply the denominators.
5 2:
6 20Ex 10 1
120 12
5 2
6 20
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Ex: x3 3x2
x2 5x 6
x2 10x 9
x2 6x 27
Step #1: Factor the numerator and the denominator.
x2 (x 3)
(x 6)(x 1)
(x 1)(x 9)
(x 9)(x 3)
Next
Ex. 5
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Step #2: Divide the numerator and denominator by the common factors.
x2 (x 3)
(x 6)(x 1)(x 1)(x 9)
(x 9)(x 3)1
1
1
1
1
1
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Step #3: Multiply the numerator and the denominator.
x2
x 6
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Multiply by the reciprocal of the divisor.
4
5
16
25
4
5
25
16
4 25
516
1
1
5
4
5
4
Remember how to divide fractions?
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Dividing rational expressions uses the same procedure.
Ex. 6: Simplify
y 2
y2 10 y 24
y2 2y
y2 2y 8
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y 2
y2 10 y 24
y2 2y
y2 2y 8
y 2
y2 10 y 24
y2 2y 8
y2 2y
y 2
(y 12)(y 2)(y 4)(y 2)
y(y 2)
1 1
1 1
4
( 12)
y
y y
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Do HW#6 Problems 1 – 10
FYI: All problems were taken from college math placement tests
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Additional Examples:
x 3
x2 4x 12
2x2 6x
x 2
Ex. 7
Ex. 8
Ex. 9
1) x + 3
2x3 2x2
x2 7x 6
x2 10x 21
2) 3x 67x 7
5x 1014x 14
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Answers:
1) x 6
2x2 (x 7)
2) 6(x 1)5(x 1)
Answer: 1
2x(x 6)