5.7: Fundamental Theorem of Algebra Assignment: P. 383-386: 1, 2, 3-33 M3,

5.7: Fundamental Theorem of Algebra
1.
2.
3.
4.
Objectives:
To apply the Fundamental
Theorem of Algebra and
its Corollary
To determine the behavior
of the graph of a function
near its zeros
To apply the Conjugates
Theorems
To use Descartes Rule of
Signs to determine the
number of +/- real zeros
Assignment:
• P. 383-386: 1, 2, 3-33 M3,
34-40 even, 50, 53-56, 58,
65-67
• Descartes and
Reflections: 1-4, 5-15 odd
• Graphical Behavior Near
Zeros: 1, 3-6
You will be able to apply the Fundamental
Theorem of Algebra and it’s Corollary
Objective 1
Exercise 1
1. How many solutions does the equation
x4 + 8x2 – 5x + 2 = 0 have?
2. How many zeros does the function
f(x) = x3 + x2 – 3x – 3 have?
Exercise 2
Find all the zeros of the function. How many x-intercepts
does it have?
f (x) = x3 – x2 – 25x + 25
Exercise 3
Find all the zeros of the function. How many x-intercepts
does it have?
g(x) = x3 + 3x2 − 9x + 5
Degrees and Zeros
The graphs below are of second degree
polynomials.
2 zeros
2 zeros
(repeated x2)
0 real zeros =
2 imaginary
Degrees and Zeros
The graphs below are of third degree
polynomials.
3 zeros
3 zeros
(1 repeated x2)
1 real zero,
2 imaginary
Degrees and Zeros
The graphs below are of fourth degree
polynomials.
Exercise 4
1. How many zeros does a quintic
polynomial have?
2. How many x-intercepts does a quintic
polynomial have?
3. Why are the answers above not
necessarily the same?
Karl Fredrich Gauss
• 1777-1855
• German
mathematician/
Child prodigy
• Could add the
numbers 1 to 100
really fast, even
as a kid!
Fun Theorem
Fundamental Theorem of Algebra
If 𝑓(𝑥) is a polynomial function of degree 𝑛,
where 𝑛 > 0, then 𝑓(𝑥) = 0 has at least one
solution in the set of complex numbers.
This means that a
polynomial function has
at least one complex
zero
Also, since real numbers
are complex numbers, the
solution could be a real
one
Fun Theorem
Fundamental Theorem of Algebra
If 𝑓(𝑥) is a polynomial function of degree 𝑛,
where 𝑛 > 0, then 𝑓(𝑥) = 0 has at least one
solution in the set of complex numbers.
– First proven by Gauss after several unsuccessful
attempts by numerous, famous mathematicians
– Perhaps more useful than this Fun Theorem is its
Corollary
Fun Corollary
Fundamental Theorem of Algebra Corollary
If 𝑓(𝑥) is a polynomial function of degree 𝑛,
then 𝑓(𝑥) = 0 has exactly 𝑛 solutions provided
that each solution repeated 𝑘 times is counted
as 𝑘 solutions.
This means that an
𝑛th degree
polynomial has 𝑛
solutions
• When a solution is repeated
𝑘 times, that solution is said
to be a repeated root with
a multiplicity of 𝑘.
Fun Theorem (Lite)
A consequent of the
Fundamental Theorem of
Algebra and it’s Corollary is
that:
y  x 2 ( x  3)( x  5)3
x0
x0
An 𝑛th degree polynomial
has 𝑛 zeros.
Sometimes at least one of these zeros
repeats 𝑘 times and is said to be a
repeated root with a multiplicity of 𝑘.
x3
x  5
x  5
x  5
6 total zeros, but only 3
x-intercepts (some repeat)
You will be able to
determine the behavior
of the graph of a function
near its zeros
Exercise 5
Determine the degree of the following polynomial
functions. How many zeros does each have? Use
a graphing calculator to determine how the
multiplicity of each zero affects the graph.
1. f(x) = (x + 5)(x – 1)2
2.
g(x) = (x + 5)3(x – 1)4
3.
h(x) = (x + 5)5(x – 1)6
Behavior Near Zeros
Real Zeros:
• Only real zeros are x-intercepts.
Imaginary zeros do not touch
the x-axis.
Odd Multiplicity:
• A zero of odd multiplicity
crosses the x-axis at that zero.
Even Multiplicity:
Odd
Multiplicity
• A zero of even multiplicity is
tangent to the x-axis at that
3
x

1


zero.
Even
Multiplicity
 x  2
2
Fun Theorem (Lite)
A consequent of the
Fundamental Theorem of
Algebra and it’s Corollary is
that:
y  x 2 ( x  3)( x  5)3
x0
x0
An 𝑛th degree polynomial
has 𝑛 zeros.
Sometimes at least one of these zeros
repeats 𝑘 times and is said to be a
repeated root with a multiplicity of 𝑘.
x3
x  5
x  5
x  5
6 total zeros, but only 3
x-intercepts (some repeat)
You will be able to apply
the Conjugates Theorems
Exercise 6a
Find all the zeros of each function. Is there any
relationship between the zeros?
f(x) = x3 + 3x2 – 14x – 20
Exercise 6b
Find all the zeros of each function. Is there any
relationship between the zeros?
g(x) = x3 + 3x2 + 16x + 130
Conjugates Theorem 1
Complex Conjugates
Theorem
If 𝑓 is a polynomial
function with real
coefficients, and
𝑎 + 𝑏𝑖 is a zero of
𝑓, then 𝑎– 𝑏𝑖 is
also a zero of f.
This means imaginary
solutions always come
in conjugate pairs.
• You have to use the
Quadratic formula (or
Completing the
Square) to find them.
Conjugates Theorem 1I
Irrational Conjugates
Theorem
If 𝑓 is a polynomial
function with real
coefficients, and 𝑎 and 𝑏
are rational numbers
such that 𝑎 + 𝑏 is an
irrational zero of 𝑓, then
𝑎– 𝑏 is also a zero of 𝑓.
• These also must be
found using the
Quadratic Formula
(or by Completing
the Square).
This means
irrational solutions
always come in
conjugate pairs.
Exercise 7
Use the Complex Conjugates Theorem to
explain why a polynomial function (with
real coefficients) of odd degree must
always have at least one real root.
Exercise 8
Write a polynomial function f of least degree that
has rational coefficients, a leading coefficient of
1, and 2 and −2 – 5i as zeros.
𝑥=2
𝑥 = −2 ± 5𝑖
𝑥 + 2 = ±5𝑖
𝑥−2=0
𝑥+2
2
= ±5𝑖
2
𝑥 2 + 4𝑥 + 4 = 25𝑖 2 = −25
𝑥 2 + 4𝑥 + 29 = 0
𝑓 𝑥 = 𝑥 − 2 𝑥 2 + 4𝑥 + 29
𝑓 𝑥 = 𝑥 3 + 2𝑥 2 + 21𝑥 − 58
Exercise 9
Write a polynomial function f of least degree
that has rational coefficients, a leading
coefficient of 1, and the given zeros.
1. −1, 2, 4
2. 4, 1 + 5
3. 2, 2i, 4 − 6
Exercise 9
Write a polynomial function f of least degree that has
rational coefficients, a leading coefficient of 1, and
the given zeros.
1. −1, 2, 4
Exercise 9
Write a polynomial function f of least degree that has
rational coefficients, a leading coefficient of 1, and
the given zeros.
1. −1, 2, 4; goes through (1, 3)
Exercise 9
Write a polynomial function f of least degree that has
rational coefficients, a leading coefficient of 1, and
the given zeros.
2. 4, 1 + 5
Exercise 9
Write a polynomial function f of least degree that has
rational coefficients, a leading coefficient of 1, and
the given zeros.
3. 2, 2i, 4 − 6
Objective 4
You will be able
to use Descartes
Rule of Signs to
determine the
number of +/real zeros
Rene Descartes
•
•
•
•
1596-1650
French philosopher-etc.
Cogito Ergo Sum
A fly taught him about
the Cartesian coordinate
plane and analytic
geometry, for which he
took full credit
Descartes Rule of Signs
Let f(x) = anxn + an – 1xn – 1 + … + a2x2 + a1x + a0 be a
polynomial with real coefficients.
The number of positive real zeros of 𝑓 is equal to
the number of changes in sign of the coefficients of
𝑓(𝑥) or is less that this by an even number.
The number of negative real zeros of 𝑓 is equal to
the number of changes in sign of the coefficients of
𝑓(−𝑥) or is less that this by an even number.
Exercise 10
Use Descartes Rule of Signs to determine the
possible number of positive real zeros, negative
real zeros and imaginary zeros for the polynomial
shown.
𝑓 𝑥 = 2𝑥 6 − 3𝑥 2 − 𝑥 + 1
Total
(+)
(−)
i
6
2
2
0
0
2
0
2
0
2
4
4
6
2 (+) or 0 (+)
𝑓 −𝑥 = 2𝑥 6 − 3𝑥 2 + 𝑥 + 1
2 (−) or 0 (−)
Total − [ (+) + (−) ] = i
Exercise 11
Use Descartes Rule of Signs to determine the possible
number of positive real zeros, negative real zeros and
imaginary zeros for each function.
1.
f(x) = x3 + 2x – 11
Exercise 11
Use Descartes Rule of Signs to determine the possible
number of positive real zeros, negative real zeros and
imaginary zeros for each function.
2.
g(x) = 2x4 – 8x3 + 6x2 – 3x + 1
Following Descartes Rules
Use Descartes Rule of Signs to determine the possible
types of zeros:
1. Use the degree to determine the total number of zeros.
2. Count the number of sign changes in 𝑓(𝑥). This is the
possible # of (+) real zeros—or less by an even #.
3. Change the signs of the odd-powered variables, and
then count the sign changes. This is the possible # of
(−) real zeros—or less by an even #.
4. Use a table to pair up the possible (+) with the
possible (−). Subtract these from the total. This must
be the possible imaginary zeros.
5.7: Fundamental Theorem of Algebra
1.
2.
3.
4.
Objectives:
To apply the Fundamental
Theorem of Algebra and
its Corollary
To determine the behavior
of the graph of a function
near its zeros
To apply the Conjugates
Theorems
To use Descartes Rule of
Signs to determine the
number of +/- real zeros
Assignment
• P. 383-386: 1, 2, 333 M3, 34-40 even,
50, 53-56, 58, 65-67
• Descartes and
Reflections: 1-4, 515 odd
• Graphical Behavior
Near Zeros: 1, 3-6
“The sum of the numbers 1 to 100 is 5050!”