Download Use synthetic division to find rational zero

Algebra 2:
Section 6.6
Finding Rational Zeros
(Day 1)
1
Definitions
 Rational
Zeros
Rational
numbers that gives an answer
of zero when plugged into the function
Where 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
3
Examples
f(x) = 2x  9 x  53x  60
3
 Only
2
find all possible rational zeros
Factors of Constant(60):  1, 2, 3, 4  5, 6, 10, 12, 15, 20, 30, 60
Factors of Leading Coefficient(2):  1, 2
Possible Rational Zeros:  1, 2, 3, 4  5, 6, 10, 12, 15, 20, 30, 60
1 3 5 15
 , , ,
2 2 2
2
4
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
1. Find the rational zeros of
f ( x)  x3  4 x 2  11x  30
List all possible rational zeros
Factors of Constant(30):
 1, 2, 3, 5, 6, 10, 15, 30
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
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1. Find the rational zeros of
f ( x)  x3  4 x 2  11x  30
Use synthetic division to find rational zero
(Check first candidate -3)
3 1  4
3
 11 30
21  30
1 7
10
0
Factor and Solve the Polynomial
f ( x)  ( x  3)( x  7 x  10)
f ( x)  ( x  3)( x  5)( x  2)
x  3, x  5, x  2
2
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Homework
 P.362
#15 - 22 all
#24 & 26 (explain directions!)
8
ADD HOMEWORK from
6.6 Day 1 to TODAY!!!
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Algebra 2:
Section 6.6
Finding Rational Zeros
(Day 2)
10
ADD HOMEWORK from
6.6 Day 1 to TODAY!!!
11
Examples
1. Find all real zeros of
f ( x)  15 x 4  68 x3  7 x 2  24 x  4
List all possible rational zeros
 1, 2, 4
Factors of Leading Coefficient(15):  1, 3, 5, 15
Factors of Constant(-4):
1 1 1
2 2
2
4 4
4
Possible Rational Zeros:  1,  ,  ,  , 2,  ,  ,  , 4,  ,  , 
3 5 15
3 5 15
3 5 15
Narrow Down Possible Rational Zeros
2 1
Probable Rational Zeros:  ,
3 5
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1. Find all real zeros of
f ( x)  15 x 4  68 x3  7 x 2  24 x  4
Use synthetic division to find rational zero
(Check first candidate -2/3)
2
 15  68
3
 10
7
15  78
45
24
4
52  30
4
6
0
Factor, then use synthetic division again
2
3
2
f ( x)  ( x  )(15 x  78 x  45 x  6)
3
13
2
3
2
f ( x)  ( x  )(15 x  78 x  45 x  6)
3
1
15  78 45
5
3  15
15  75
6
6
30
0
Factor, then solve using quadratic and zero product
property
2
1
f ( x)  ( x  )( x  )(15 x 2  75 x  30)
3
5
2
1
x   0, x   0, 15 x 2  75 x  30  0
3
5
14
2
1
x   0, x   0, 15 x 2  75 x  30  0
3
5
2
75

75
 4(15)(30)
2
1
x , x , x
3
5
2(15)
75  3825 75  225 17
x

30
30
75  15 17

2
1
5  17
30
x , x , x
3
5
2
5  17

2
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Homework
 P.363
#48-58 evens, omit #54
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