1. Descartes Rule of Signs Positive real zeros = Negative real zeros =
2.5 cont.
Provides more specific information about the number of real zeros a polynomial can have.
The number of sign changes of f(x) or less than this by an even integer.
The number of sign changes of f(-x) or less than this by an even integer.
Descartes Rule of Signs
Positive real zeros =
Negative real zeros = 1

2. Example
Determine the possible number of positive and negative real zeros
of f(x) = x3 + 7x2 + x + 7
You try one
Determine the possible number of positive and negative real zeros
of f(x) = x3 + 2x2 + 5x + 4 2

3. Linear Factorization Theorem
Enables us to find a polynomial function when the zeros are known.
An nth-degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1. 3

4. Example
Find an nth-degree polynomial function with real coefficients satisfying the given conditions.
n = 3; 4 and 2i are zeros; f(-1) = 50 4

5. You try one
Find an nth-degree polynomial function with real coefficients satisfying the given conditions.
n = 3; -3 and i are zeros; f(1) = 8 5

6. P. 336 #17 – 37 odd 6