The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

If F(x) is a polynomial of degree n, where n>0, f has at last one zero in the complex number system

Linear Factorization Theorem

If f(x) is a polynomial of degree n where n>0, f has precisely n linear factors

f(x)=an(x-c1)(x-c2). . . (x-cn)

where C1, C2......., Cn are complex numbers

Example: Real Zeros of Polynomial Functions

a.) The first degree polynomial F(x)=x-2 has exactly one zero: x=2

b.) Countring multiplicity, the second degree polynomial function

f(x)=x^2 - 6x + 9 = (x - 3)(x -3)

has exactly two zeros: x = 3 and x= 3.

A third-degree polynomial has THREE zeros

A fourth-degree polynomial has FOUR zeros. . . ect.

Example: Finding the Zeros of a Polynomial Function

f(x) = x^5 + x^3 + 2x^2 - 12x + 8

[ (+ -) (1,2,4,8) ] All possible Rational Zeros

Use synthetic division, or look to graph of equation for help.

After using synthetic division you can determine the zeros for the equation

F(x)= (x - 1)(x - 1)(x+ 2)(x^2 + 4)

How to factor x^2 + 4

X^2 - (-4) = (X- √-4) (X- √-4) = (X - 2i) (X - 2i)

All zeros = 1, 1, -2, 2i, and -2i