1. Determine the number and type of roots for a polynomial equation
7.5 Roots and Zeros
Determine the number and type of roots for a polynomial equation

2. 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.

3. 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.

4. 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.
Count the amount of sign switches
Two switches,
either 2 positive or 0 positive zeros

5. Descartes Rule of Signs
Checking for negative roots, use the function with (-x).
Count the amount of sign switches
Two switches,
either 2 negative or 0 negative zeros

6. Descartes Rule of Signs
Making a table of type of zeros.
Number of positive zeros
Number of negative zeros
Number of imaginary zero 2 4 6

7. 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

8. 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)

9. 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

10. 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))

11. 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

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)
(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

13. Homework
Page 375 – 376
# 13, 17, 19 – 23odd,
35 , 37, 39

14. Homework
Page 375 – 376
# 14, 18, 20 – 24even,
36 , 38, 54,
58,