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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

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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