Download Fundamental Theorem of Algebra

2-5 Fundamental Theorem of Algebra
If P(x) is a polynomial function with real
coefficients of degree n>0, then there exist
complex numbers
a,c ,c , ,cn with a 0
1 2
such that
P(x)  a(x c1)(x c2) (x cn)
In other words, a polynomial with real
coefficients can be completely factored over the
complex number system into linear factors.
The complex zeros of a polynomial with real
coefficients will occur in conjugate pairs:
a bi and a bi
Example: Find all zeroes and write in factored
form.
1.
f (x)  x4 1 0
Do
f ( x)  x3  4x  0
Ex. 2
f (x)  x3  2x2  3x
Ex. 3 Find a 4th degree polynomial with real
coefficients which has zeroes –1, –1, and 3i