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 al! 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

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