Fundamental Theorem of Algebra
The fundamental theorem of Algebra tells us the maximum number
of distinct zeros (real or imaginary) a function can potentially have.
 A polynomial function with degree n has at most n distinct zeros.
o EX: A fifth degree polynomial could have 5 zeros.
o Note: Some zeros or solutions are repeated.
 EX: (x – 2)2 = 0 has only one distinct zero (x = 2).
 We would call x = 2 a double root/zero.
o Note: Imaginary zeros always come in pairs.
Examples: Find all zeros (real and imaginary) of the following functions.
1. f(x) = 3x3 – 7x - 10
2. f(x) = x4 + 3x3 - 8x2 - 22x - 24
Writing Polynomial Functions
1. Use the zeros to write the function out in factored form.
2. Multiply the factors to write the polynomial in standard form.
Examples: Write a polynomial function of least degree that has leading
coefficient of 1 and the given zeros.
1. zeros: x = -3, x = 1, x = 2, x = -2 2. zeros: x = 5, x = 3 + i