Transcript 7.5 Roots and Zeros

7.5 Roots and Zeros
Determine the number and type of
roots for a polynomial equation
Fundamental Theorem of Algebra
Every polynomial equation with degree
greater than zero has at least one root in
the set of complex number.
Polynomial of degree “n” has “n” number
of zeros.
4x4 – 5x3 + 2x – 10 = 0,
has 4 zero since it is a polynomial of
degree 4.
Complex zeros come in pairs
If a polynomial has a zero of 3 +2i , then it
has another zero of 3 – 2i.
Zero of complex roots always are in
pairs.
4i means there is another zero -4i.
The sign in front of the imagery part
changes.
Descartes Rule of Signs
To tell the type of zeros, check the number
of times the sign in front of the coefficients
change.
Checking for positive roots, use the original
function.
6
3
2
f x    x  4 x  2 x  x  1
Count the amount of sign switches
Two switches,
either 2 positive or 0 positive zeros
Descartes Rule of Signs
Checking for negative roots, use the function
with (-x).
6
3
2
f  x    x   4 x   2 x    x   1
f  x    x 6  4 x 3  2 x 2  x  1
Count the amount of sign switches
Two switches,
either 2 negative or 0 negative zeros
Descartes Rule of Signs
Making a table of type of zeros.
f x    x 6  4 x 3  2 x 2  x  1
Number of
Number of
positive zeros negative
zeros
2
2
2
0
0
2
0
0
Number of
imaginary
zero
2
4
4
6
How to write an equation given the
zeros
Given zeros of 3, 4 and -1
The zeros are answers to
(x – 3) = 0
(x – 4) = 0
( x + 1) = 0
The original polynomial (x -3)(x – 4)(x + 1)
Multiply (x2 – 7x + 12)(x + 1)
x3 – 7x2 + 12x + x2 – 7x + 12
f(x) = x3 – 6x2 + 5x + 12
How to write an equation given the
zeros
Given zeros of 2, 3i
Remember that means – 3i is also a zero
(x – 2)(x- 3i)(x + 3i)
How to write an equation given the
zeros
Given zeros of 2, 3i
Remember that means – 3i is also a zero
(x – 2)(x- 3i)(x + 3i)
(x – 2)(x2 +3xi – 3xi – 9i2)
i2 = - 1
How to write an equation given the
zeros
Given zeros of 2, 3i
Remember that means – 3i is also a zero
(x – 2)(x- 3i)(x + 3i)
(x – 2)(x2 +3xi – 3xi – 9i2)
i2 = - 1
(x – 2)(x2 – 9(-1))
How to write an equation given the
zeros
Given zeros of 2, 3i
Remember that means – 3i is also a zero
(x – 2)(x- 3i)(x + 3i)
(x – 2)(x2 +3xi – 3xi – 9i2)
i2 = - 1
(x – 2)(x2 – 9(-1))
(x – 2)(x2 + 9) = x3 – 2x2 + 9x – 18
How to write an equation given the
zeros
Given zeros of 2, 3i
Remember that means – 3i is also a zero
(x – 2)(x- 3i)(x + 3i)
(x – 2)(x2 +3xi – 3xi – 9i2)
i2 = - 1
(x – 2)(x2 – 9(-1))
(x – 2)(x2 + 9) = x3 – 2x2 + 9x – 18
f(x) = x3 – 2x2 + 9x – 18
Homework
Page 375 – 376
# 13, 17, 19 – 23odd,
35 , 37, 39
Homework
Page 375 – 376
# 14, 18, 20 – 24even,
36 , 38, 54,
58, 60 - 62