2-5 Fundamental Theorem of Algebra

2-5 Fundamental Theorem of Algebra
You have been using the fact that an nth degree polynomial can
have at most n real zeros.
In the complex number system every nth degree polynomial
function has precisely n zeros.
Fundamental Theorem of Algebra - If f(x) is a polynomial of
degree n, where n>0, then f has at least one zero in the complex
system.
Linear factorization theorem - If f(x) is polynomial of degree n
then f has precisely n linear factors.
Ex. 1 a) f(x) = x - 6
has one zero : x = 6
b) f(x) = x 2 - 6x + 9 =
(x-3)(x-3)
has two repeated zeros: x = 3, x = 3 multiplicity of 2
c) f(x) = x 3 + 4x =
x (x 2 + 4) = x(x - 2i)(x + 2i)
has three zeros at x = 0, x = 2i, x = -2i
d) f(x) = x 4 - 1 =
(x 2 + 1)(x 2 - 1) =
(x 2 + 1)(x + 1)(x - 1)= (x + i)(x - i)(x + 1)(x - 1)
has four zeros at x = -i, x = i, x = -1, x = 1
1

Ex. 2 Find all the zeros of the function and write the polynomial
as a product of linear factors.
f(x) = x 2 -12x+26
2

Ex. 3 f(x) = x 5 + x 3 + 2x 2 - 12x + 8 List all zeros.
n =
n - 1 =
L:
R:
+:
-:
Possible rational zeros:
Sketch the graph:
3

Conjugate pairs: a + bi and a - bi
Complex numbers occur in conjugates pairs
Let f(x) be a polynomial function that has real coefficients.
If a + bi, where b does not equal 0, is a zero of the function,
the conjugate pair a - bi is also a zero of the function.
Ex. 4 Find a 4th degree polynomial function with real coefficients,
that has 1, -1, and 3i as zeros.
f(x) = (x+1)(x-1)(x-3i)(x+3i)=
4

ASSN: p. 144,145 1-5 odd,
9,13,15,19,23,29,31,37,39,43,47,
51,
5