Section 2-5 Zeros of Polynomial Functions.notebook

Section 2­5 Zeros of Polynomial Functions.notebook
September 23, 2014
Properties of Polynomial Equations
1. If a polynomial equation is of degree n, then counting multiple
roots separately, the equation has n roots.
2. If a + bi is a root of a polynomial equation, then the complex imaginary number a ­ bi is also a root. Complex imaginary
roots, if they exist, occur in conjugate pairs. Fundamental Theorem of Algebra
A polynomial with highest degree n has exactly n roots and n factors. (These can be real or imaginary roots!) f(x) = 3x3 ­ 2x2 + 7 ⇒
g(x) = 5x6 ­ 3x ­ 2 ⇒
h(x) = 3x­2 ­ 3x­1 + 1 ⇒
zero's nee
d not be all re
Linear Factorization Theorem
Given f(x) = a0xn + a1xn­1 + a2xn­2 +...+ an, and c1, c2, c3,... cn are zeros of f(x), then f(x) can be expressed by the product of linear factors. f(x) = (x ­ c1)(x­c2)(x­c3)...(x­cn)
Find a 3rd degree polynomial function f(x) with real coefficients that has ­3 and i as zeros and such that f(1) = 8