Finding Zeros of Polynomials

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Finding Zeros of Polynomials
Last updated 12-4-07
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Divide
- ( )
- ( )
- ( )
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Divide
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Use Synthetic Division to Divide
x 2 0
1 4 -5 3
-2 -2 -4 18
1 2 -9 21
x -2
remainder
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Use Synthetic Division to Divide
x - 3 0
2 -11 3 36
3 6 -15 -36
2 -5 -12 0
x 3
Factored Form
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Factor
x 1 0
1 4 -15 -18
-1 -1 -3 18
1 3 -18 0
x -1
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Find the zeros of
x 3
x 3 0
6 -7 -43 30
3 18 33 -30
6 11 -10 0
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Factor Theorem
A polynomial f(x) has a factor x kif and only
if f(k) 0.
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Rational Zero Theorem
If f(x) anxn . . . a1x a0 has integer
coefficients, then every rational zero of f(x)
has the following form
p factor of constant term a0
q factor of leading coefficient an
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List the possible rational zeros
1, 3, 5, 15
p factor of constant term a0
q factor of leading coefficient an
15
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1, 2, 3, 6
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List the possible rational zeros
1, 2, 3, 4, 6, 8, 12, 24
p factor of constant term a0
q factor of leading coefficient an
24
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1, 3, 9
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Find all real zeros
1, 3
p factor of constant term a0
q factor of leading coefficient an
3
8
1, 2, 4, 8
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Find all real zeros
x 1
8 2 -21 -7 3
1 8 10 -11 -18
8 10 -11 -18 -15
Remainder ? 0
Therefore, not a factor.
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Find all real zeros
x 3
8 2 -21 -7 3
3 24 78 171 492
8 26 57 164 495
Remainder ? 0
Therefore, not a factor.
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Find all real zeros
8 2 -21 -7 3
-12 15 9 -3
8 -10 -6 2 0
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Find all real zeros
3 1, 3
8 1, 2, 4, 8
1
4
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Find all real zeros
4 -5 -3 1
1 -1 -1
4 -4 -4 0
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Find all real zeros
Rational
Rational
Irrational but Real
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Look at the graph
1.618 -0.618
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Look at the graph
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Find all real zeros
1, 5, 25
p factor of constant term a0
q factor of leading coefficient an
25
1
1
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Find all real zeros
x 1
1 0 -16 -40 -25
1 1 1 -15 -55
1 1 -15 -55 -80
5 5 25 45 25
1 5 9 5 0
x 5
x 5 0
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Find all real zeros
x -1
1 5 9 5
-1 -1 -4 -5
1 4 5 0
x 1 0
24
Find all real zeros
Imaginary -- not Real
25
Look at the graph
Note x-min -10x-max 10x-scale 1 y-min
-250y-max 100y-scale 50
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Find all real zeros
1, 2, 3, 4, 6, 12
p factor of constant term a0
q factor of leading coefficient an
12
1
1
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Find all real zeros
x 1
1 -3 -5 15 4 -12
1 1 -2 -7 8 12
1 -2 -7 8 12 0
x - 1 0
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Find all real zeros
x 2
1 -2 -7 8 12
2 2 0 -14 -12
1 0 -7 -6 0
x - 2 0
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Find all real zeros
x 3
1 0 -7 -6
3 3 9 6
1 3 2 0
x - 3 0
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Find all real zeros
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Look at the graph
End Behavior?
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Look at the graph
Note x-min -5x-max 5x-scale 1 y-min
-20y-max 20y-scale 5
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Fundamental Theorem of Algebra
If f(x) is a polynomial function of degree n
where ngt0, then the equation f(x) 0 has at
least one solution in the set of complex numbers.
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Corollary to the Fundamental Theorem of Algebra
If f(x) is a polynomial function of degree n
where ngt0, then the equation f(x) 0 has exactly
n solutions provided each solution repeated twice
is counted as 2 solutions, each solution repeated
three times is counted as 3 solutions, and so on.
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Find all real zeros
1, 3, 9
p factor of constant term a0
q factor of leading coefficient an
9
1
1
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Find all real zeros
1 is a multiple root with multiplicity 3
-3 is a multiple root with multiplicity 2
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Find all real zeros
1, 2, 4, 5, 10, 20
p factor of constant term a0
q factor of leading coefficient an
20
1
1
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Find all real zeros
1 -6 7 16 -18 -20
1 1 -5 2 18 0
1 -5 2 18 0 -20
2 2 -8 -2 28 20
1 -4 -1 14 10 0
x 1
x 2
x - 2 0
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Find all real zeros
1 -4 -1 14 10
2 2 -4 -10 8
1 -2 -5 4 18
-1 -1 5 -4 -10
1 -5 4 10 0
x 2
x -1
x 1 0
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Find all real zeros
1 -5 4 10
-1 -1 6 -10
1 -6 10 0
x -1
x 1 0
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Find all real zeros
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End Behavior?
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Key Concepts
If f(x) is a polynomial function with real
coefficients, and a bi is an imaginary zero of
f(x), then a - bi is also a zero of f(x).
Imaginary solutions appear in conjugate pairs.
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Key Concepts
If f(x) is a polynomial function with rational
coefficients, and a and b are rational numbers
such that ---- is irrational. If --------- is a
zero of f(x), then --------- is also a zero of
f(x).
Irrational solutions containing a square root
appear in conjugate pairs.
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Write a polynomial function f(x) of least degree
that has rational coefficients, leading
coefficient of 1, and the following zeros ? 1,
-2, 4.
x 1
x -2
x 4
x 1 0
x 2 0
x 4 0
f(x) (x 1) (x 2) (x 4)
f(x) (x 1) (x2 4x 2x 8)
f(x) (x 1) (x2 2x 8)
f(x) x3 2x2 8x x2 2x 8
f(x) x3 3x2 6x 8
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Write a polynomial function f(x) of least degree
that has rational coefficients, leading
coefficient of 1, and the following zeros ?
WAIT !
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Write a polynomial function f(x) of least degree
that has rational coefficients, leading
coefficient of 1, and the following zeros ?
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(No Transcript)
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Write a polynomial function f(x) of least degree
that has rational coefficients, leading
coefficient of 1, and the following zeros ?
WAIT !
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Write a polynomial function f(x) of least degree
that has rational coefficients, leading
coefficient of 1, and the following zeros ?
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(No Transcript)
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(No Transcript)
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Descartes Rule of Signs
Let f(x) anxn an-1xn-1 . . . a1x a0 be a
polynomial function with real coefficients. ? The
number of positive real zeros of f(x) is equal to
the number of changes in sign of the coefficients
of f(x) or is less than this by an even number. ?
The number of negative real zeros of f(x) is
equal to the number of changes in sign of the
coefficients of f(-x) or is less than this by an
even number.
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Determine the number of positive real zeros.
1
2
3
3 sign changes ? f(x) has 3 or 1 positive
real zero(s).
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Determine the number of negative real zeros.
1
2
2 sign changes ? f(x) has 2 or 0 negative
real zero(s).
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Putting it together !
? f(x) has 3 or 1 positive real zero(s)
? f(x) has 2 or 0 negative real zero(s)
Positivereal zeros Negativereal zeros Imaginaryzeros Total zeros
3 2 0 5
3 0 2 5
1 2 2 5
1 0 4 5
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Look at the graph
Note x-min -5x-max 5x-scale 1 y-min
-30y-max 20y-scale 5
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Determine the number of positive real zeros.
0 sign changes ? f(x) has 0 positive real
zero(s).
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Determine the number of negative real zeros.
1
2
3
3 sign changes ? f(x) has 3 or 1 negative
real zero(s).
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Putting it together !
? f(x) has 0 positive real zero(s)
? f(x) has 3 or 1 negative real zero(s)
Positivereal zeros Negativereal zeros Imaginaryzeros Total zeros
0 3 4 7
0 1 6 7


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Look at the graph
Note x-min -5x-max 5x-scale 1 y-min
-50y-max 50y-scale 10
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Find all real zeros
1, 3, 7, 21
p factor of constant term a0
q factor of leading coefficient an
21
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1, 2, 3, 6, 9, 18
? f(x) has 2 or 0 positive real zero(s)
? f(x) has 1 negative real zero(s)
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Find all real zeros
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Find all real zeros
18 -63 40 21
1 18 -45 -5
18 -45 -5 16
2 36 -54 -28
18 -27 -14 -7
3 54 -27 39
18 -9 13 60
7 126 441 3367
18 63 481 3388
upper bound
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Find all real zeros
X
X
X
X
X
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Find all real zeros
Location Principle
18 -63 40 21
1 18 -45 -5
18 -45 -5 16
2 36 -54 -28
18 -27 -14 -7
3 54 -27 39
18 -9 13 60
7 126 441 3367
18 63 481 3388
upper bound
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Find all real zeros
X
X
X
X
X
X
X
X
X
X
X
X
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Find all real zeros
18 -63 40 21
27 -54 -21
18 -36 -14 0
42 14
18 6 0