Download 2.5 Zeros of Polynomial Functions

2.5 Zeros of Polynomial
Functions
Fundamental Theorem of Algebra
Rational Zero Test
Upper and Lower bound Rule
Fundamental Theorem of Algebra
If f(x) is a polynomial of degree “n” > 0, then
f(x) has at least one zero in the complex
number system.
Complex zero’s (roots) come in pairs
If a + bi is a zero, then a – bi is a zero.
Linear Factorization Theorem
If f(x) is a polynomial of degree “n”>0, then there
are as many zeros as degree.
If f(x) is a third degree function, then
f(x) = an(x – c1)(x – c2)(x – c3) where c are complex
numbers.
Complex zero’s (roots) come in pairs
If a + bi is a zero, then a – bi is a zero.
The Rational Zero Test
If f(x) has integer coefficients, then all
possible zeros are
factors of the constant
factor of the lead coefficient
The Rational Zero Test
If f(x) has integer coefficients, then all
possible zeros are
factors of the constant
factor of the lead coefficient
f(x) = x 3 – 7x 2 + 4x + 12
Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12
±1
f(x) = x 3 – 7x 2 + 4x + 12
Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12
±1
-1|1
-7 4
12
-1 8 -12
1 - 8 12 0
So – 1 is a zero
How do you want to find the other zeros.
x 2 – 8x + 12
Find the zeros
f(x) = 3x3 – x2 + 6x - 2
Descartes' Rule of Signs
Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with
real coefficients and a0 ≠ 0.
Part 1
The number of positive real zeros equals (or
a even number less), the number of
variation in the sign of the coefficient
(switching from positive to negative or
negative to positive).
Descartes' Rule of Signs
Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with
real coefficients and a0 ≠ 0.
Part 2
The number of negative real zeros equals
(or a even number less), the number of
variation in the sign of the coefficient
(switching from positive to negative or
negative to positive) in f(- x).
Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
How many times does the sign change ?
Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
How many times does the sign change ?
3 times.
There are 3 or 1 positive zeros.
Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
What about f( -x) = -4x3 – 3x2 – 2x - 1
How many times does the sign change ?
Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
What about f( -x) = -4x3 – 3x2 – 2x - 1
How many times does the sign change ?
No change, no negative zeros.
Upper and Lower bound Rule
If c > 0 ( “c” the number you divide by) and the last
row of synthetic division is all positive or zero,
the c| is the upper bound
So there is no zero larger then c, where c > 0.
If c < 0 and the last row alternate signs
( zero count either way), then c is the lower bound.
f(x) = 2x3 – 5x2 + 12x - 5
Check to see if 3 is the upper bound?
3| 2
- 5 12 - 5 All signs are
6
3 45
positive.
2
1 15 40
3 is an upper bound
f(x) = 2x3 – 5x2 + 12x - 5
Check to see if - 1 is the lower bound?
- 1| 2
- 5 12 - 5 All signs are
-2
7 -19
switch.
2
- 7 19 -24
-1 is an lower bound
f(x) = 2x3 – 5x2 + 12x - 5
Find the zeros
Homework
Page 160 – 164
# 5, 15, 23, 35,
42, 50, 57, 65,
73, 81, 85, 93,
103, 108, 111
Homework
Page 160 – 164
# 9, 19, 29, 41,
53, 61, 64, 77,
87, 97, 105,125
One more time
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