Cloud Frontiers: A Deep Dive into Serverless Spatial Data and FME
Ca8e Ppt 5 6
1. Section 5.6
Complex Zeros;
Fundamental Theorem of
Algebra
2.
3.
4.
5.
6. Example 1
A polynomial of degree 5 whose coefficients are real numbers
has the zeros -2, -3i, and 2+4i. Find the remaining two zeros.
7. By the conjugate pairs theorem, if x = -3i and x = 2+4i
then x = 3i and x = 2-4i
8. Example 2: Find the remaining zeros of f.
Degree 5; zeros: 1, i, 2i
The degree
We have three
indicates the
zeros, so we are
number of zeros a
missing two!!!
polynomial has.
By the Conjugate
Pairs Theorem Remaining zeros: -i, -2i
9.
10. Example 3
Find a polynomial f of degree 4 whose coefficients are
real numbers and that has the zeros 1, 1, 4+i.
11. By the Conjugate
Pairs Theorem
x = 1, x = 1, x = 4 + i, x = 4 − i
P ( x) = ( x − 1)( x − 1)( x − 4 − i )( x − 4 + i )
P( x) = ( x − x − x + 1)( x − 4 x + xi − 4 x +
2 2
16 − 4i − xi + 4i − i ) 2
P( x) = ( x − 2 x + 1)( x − 8 x + 16 − (−1))
2 2
P( x) = ( x − 2 x + 1)( x − 8 x + 17)
2 2
12. P( x) = x − 8 x + 17 x − 2 x + 16 x −
4 3 2 3 2
34 x + x − 8 x + 17
2
P( x) = x − 10 x + 34 x − 42 x + 17
4 3 2
13.
14.
15. Example 4: Use the given zero to find the remaining
zeros of each function.
f ( x) = x − 4 x + 4 x − 16; zero : 2i
3 2
2i 1 -4 4 - 16
2i - 4 − 8i 16
1 - 4 + 2i - 8i 0
Note:
2i (-4 + 2i ) = -8i + 4i = -8i + 4(-1) = -4 − 8i
2
2i (-8i ) = -16i = -16(-1) = 16
2
17. Example 5
Find the complex zeros of the polynomial function
x 4 + 2 x3 + x 2 − 8 x − 20 ±1, ± 2, ± 4, ± 5, ± 10, ± 20
18. 2 1 2 1 -8 - 20
2 8 18 20
1 4 9 10 0
-2 1 4 9 10
-2 -4 - 10
1 2 5 0
x + 2x + 5 = 0
2 +5
Note: The resulting quadratic equation can not be
factored since there is no number that multiplied gives
you five and at the same time added gives you two.
19. We have to use completing the square or the quadratic
formula 2
x + 2x + 5 = 0
a =1 b=2 c=5
Quadratic Formula
− b ± b − 4ac
2
x=
2a
Substitute − (2) ± (2) − 4(1)(5)
2
x=
2(1)
20. Simplify and solve!
− 2 ± 4 − 20
x=
2
− 2 ± − 16
x=
2
− 2 ± 4i
x= = −1 ± 2i
2
Complex zeros: x = −1± 2i
Real zeros: x = ±2
Remember the problem only asks for the complex zeros!
