1 algebra 2: section 6.6 finding rational zeros (day 1)
TRANSCRIPT
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Algebra 2: Section 6.6
Finding Rational Zeros
(Day 1)
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Definitions
Rational ZerosRational numbers that gives an answer
of zero when plugged into the functionWhere the graph of the function
crosses the x-axis
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The Rational Zero Theorem
If a function has integer coefficients, then every rational zero is in this form……division:
a factor of the constant term
a factor of the leading coefficient
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Examples
Only find all possible rational zeros
Possible Rational Zeros: 1, 2, 3, 4 5, 6, 10, 12, 15, 20, 30, 60
1 3 5 15, , ,
2 2 2 2
3 2f(x) = 2x 9 53 60x x
Factors of Constant(60): 1, 2, 3, 4 5, 6, 10, 12, 15, 20, 30, 60
Factors of Leading Coefficient(2): 1, 2
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Steps to find rational zeros of a function
List all the possible rational zeros Narrow down possible rational zeros using
graphing calculator Test the possible rational zeros using
synthetic division “Factor the polynomial and solve”
and/or “Use the quadratic formula” and/or “Continue to test more possible
zeros using synthetic division”…(tomorrow)
Solve
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Examples
3 2
1. Find the rational zeros of
( ) 4 11 30f x x x x List all possible rational zeros
Factors of Leading Coefficient(1): 1
Possible Rational Zeros: 1, 2, 3, 5, 6, 10, 15, 30
Narrow Down Possible Rational Zeros
Probable Rational Zeros: 3, 2, 5
Factors of Constant(30): 1, 2, 3, 5, 6, 10, 15, 30
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3 2
1. Find the rational zeros of
( ) 4 11 30f x x x x
1 4 11 30 3
Use synthetic division to find rational zero(Check first candidate -3)
Factor and Solve the Polynomial2( ) ( 3)( 7 10)f x x x x
( ) ( 3)( 5)( 2)f x x x x
3 21 30
1 7 10 0
3, 5, 2x x x
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Homework
P.362#15 - 22 all#24 & 26 (explain directions!)
ADD HOMEWORK from
6.6 Day 1 to TODAY!!!
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Algebra 2: Section 6.6
Finding Rational Zeros
(Day 2)
ADD HOMEWORK from
6.6 Day 1 to TODAY!!!
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Examples
4 3 2
1. Find all real zeros of
( ) 15 68 7 24 4f x x x x x List all possible rational zeros
1 1 1 2 2 2 4 4 4Possible Rational Zeros: 1, , , , 2, , , , 4, , ,
3 5 15 3 5 15 3 5 15
Narrow Down Possible Rational Zeros
2 1Probable Rational Zeros: ,
3 5
Factors of Constant(-4): 1, 2, 4 Factors of Leading Coefficient(15): 1, 3, 5, 15
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Use synthetic division to find rational zero(Check first candidate -2/3)
Factor, then use synthetic division again
3 22( ) ( )(15 78 45 6)
3f x x x x x
15 68 7 24 4
10 52 30 4
15 78 45 6 0
4 3 2
1. Find all real zeros of
( ) 15 68 7 24 4f x x x x x
15 68 7 24 4
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3 22( ) ( )(15 78 45 6)
3f x x x x x
3 15 6
15 75 30 0
1
515 78 45 6
Factor, then solve using quadratic and zero product property
22 1( ) ( )( )(15 75 30)
3 5f x x x x x
22 10, 0, 15 75 30 0
3 5x x x x
15
275 75 4(15)(30)
2(15)
2 , 1 ,53
xx x
75 3825
30x
75 225 17
30
75 15 17
30
5 17
2
2 1 5 17, ,
3 5 2x x x
22 10, 0, 15 75 30 0
3 5x x x x
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Homework
P.363#48-58 evens, omit #54