Transcript 5.7: Fundamental Theorem of Algebra
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
Vocabulary
Theorem Real Numbers Irrational Conjugates Corollary Complex Numbers Complex Conjugates
Exercise 1
1.
How many solutions does the equation
x
4 + 8
x
2 – 5
x
+ 2 = 0 have?
2.
f
How many zeros does the function (
x
) =
x
3 +
x
2 – 3
x
– 3 have?
Exercise 2
Determine the degree of the polynomial functions. How many zeros does each have? Use a graphing calculator to graph each function. Find all the zeros of each function.
1.
f
(
x
) =
x
3 –
x
2 – 25
x
+ 25 2.
g
(
x
) =
x
3 + 7
x
2 + 12
x
+ 10
Exercise 3
Determine the degree of the polynomial functions. How many zeros does each have? Use a graphing calculator to graph each function. How many
x
-intercepts does each have?
1.
f
(
x
) =
x
3 –
x
2 – 25
x
+ 25 2.
g
(
x
) =
x
3 + 7
x
2 + 12
x
+ 10
Degrees and Zeros
The graphs below are of
second
degree polynomials.
2 zeros 1 zero, repeated x2 0 real zeros = 2 imaginary
Degrees and Zeros
The graphs below are of
third
degree polynomials.
3 zeros 2 zeros, 1 repeated x2 1 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
n
th 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
2 ( 3)(
x
5) 3 An 𝑛 th has 𝑛 degree polynomial zeros.
x x
0 0
x
3
x x x
5 5 5 Sometimes at least one of these zeros repeats 𝑘 times and is said to be a repeated root with a
multiplicity
of 𝑘 .
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:
A zero of even multiplicity is
tangent
zero.
to the
x
-axis at that Odd Multiplicity
x
1 3 Even Multiplicity
x
2 2
You will be able to apply the Conjugates Theorems
Exercise 6
Find all the zeros of each function. Is there any relationship between the zeros?
1.
f
(
x
) =
x
3 + 3
x
2 – 14
x
– 20 2.
g
(
x
) =
x
3 + 3
x
2 + 16
x
+ 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 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 – 5
i
as zeros.
x
2
x
i x
2 0
x
x
2 5
i
2
x
2 4
x
25
x
2 4
x
29 0
i
2 25
x
2
x
2 4
x
29
x
3 2
x
2 21
x
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, 2
i
, 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
) =
a n x n
+
a n
– 1
x n
– 1 + … +
a
2
x
2 polynomial with real coefficients.
+
a
1
x
+
a
0 be a 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.
2
x
6 3
x
2 1
Total
f
( 2 (+) or 0 (+) 2
x
6 3
x
2 1 6
(+)
2 2 0 0
( −)
2 0 2 0 2 4 4 6
i
2 ( − ) or 0 ( − ) Total − [ (+) + (−) ] =
i
Following Descartes Rules
Use Descartes Rule of Signs to determine the possible types of zeros: 1.
2.
Use the degree to determine the total number of zeros.
Count the number of sign changes in 𝑓(𝑥) . This is the possible # of (+) real zeros —or less by an even #.
3.
4.
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 #.
Use a table to pair up the possible (+) with the possible ( − ). Subtract these from the total. This must be the possible imaginary zeros.
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
) =
x
3 + 2
x
– 11 2.
g
(
x
) = 2
x
4 – 8
x
3 + 6
x
2 – 3
x
+ 1
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