Download Today you will Write a polynomial equation of least degree for a

Today you will
Write a polynomial equation of least degree
for a given set of roots.
4.1
Polynomial Functions
Which are polynomials in one variable?
6
2
4
5x - 8x + 7x 3- 4.3x + 19
6
-4
3
2
5x - 8x + 7x - 4.3x + 19
6
2
5x - 8ix4 + 7x3 - 4.3x + 19
Text pg 209 # 5-8, 15-20
State the degree and leading coefficient of each polynomial
5.
a 3 + 6a +14
6.
5m2 + 8m5 - 2
Determine whether each number is a root of
x3 - 5x2 - 3x - 18 = 0
7.
5
8.
6
State the degree and leading coefficient of each polynomial
15.
4
3
16.
5t + t - 7
2
3
5
3
17. 9a + 5a - 10
19.
6
p + 7p - p
18.
20.
3x 7 - 4x5 + x 3
5
14b - 25b
14y + 30 + y
2
Roots/Zeros/Solutions
A root or zero may also be an
imaginary number, such as 3i.
To review, the imaginary unit i =
-1
2
i = -1
3
i = i 2 i = -i
4
i = i2 i 2= 1
Complex numbers are the set of imaginary numbers
combinted with the set of real numbers.
A complex number is any number in the form of a + bi
(a and b are real numbers)
If b = 0, then the complex number is a real number.
If a = 0 and b = 0 then the complex number is called
a pure imaginary number.
The Fundamental Theorem of Algebra
**Every polynomial equation with degree
greater than zero has at least one root in
the set of complex numbers.
Corollary to the Fundamental Theorem of
Algebra
**Every polynomial of degree n (n>0) can be
written as the product of a constant k (k=0)
and n linear factors.
A polynomial equation of degree n has
exactly n complex roots.
The graph of a polynomial function with odd degree must
cross the x-axis at least once.
The graph of a polynomial function with even degree
may or may not cross the x-axis.
If it does it will cross an even number of times.
Each x-intercept represents a real root of the
polynomial equation.
Write a polynomial equation of least degree for each set of
roots.
Does the equation have an odd or even degree?
How many times does the graph of the related equation
cross the x-axis?
Example 1:
Roots are
-5 and 7
Example 2:
Roots are
2, 4i, and -4i
Example 3:
Roots are
6, 2i, -2i, i, and -i
HW:
4.1 (pg 210)
# 22, 29-38