Section+2.5

Section 2.5

Theorems to know...


 * The Fundamental Theroem of Algebra**- A polynomial function of degree n has n complex zeros(real and nonreal). Some of these zeros may be repeated.

f(x) = a(x-z1)(x-z2)...(x-zn) where a is the leading coefficient of f(x) and z1, z2....zn are the complex zeros of f(x). The zi are not necessarily distinct numbers; some may be repeated.
 * Linear Factorization Theorem**- If F(x) is a polynomial function of degree n>0 then F(x) has precisely n linear factors and


 * The following statements about a polynomial function f are equivalent if k is a complex number:**

1. x=k is a solution of the equation f(x)=0. 2. k is a zero of the function f. 3. x-k is a factor of f(x)


 * Complex Conjugate Zeros**- Suppose that f(x) is a polynomial function with real //coefficients//. If a and b are real numbers with b not = to 0 and a=bi is a zero of f(x), then its complex conjugate a-bi is also a zero of f(x).


 * Factors of a Polynomial with Real Coefficients**- Every polynomial with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.


 * Polynomial Function of Odd Degree**- Every polynomial function of odd degree with real coefficients has at least one real zero.