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 of degree n with real
coefficients can be completely factored over
the complex number system into n linear
factors
 a polynomial of degree n with real
coefficients will have exactly n complex
zeros
Finding Zeros:
A. The Rational Zero Test
The rational zero test is a way of finding the possible
rational zeros of a polynomial by using the lead
coefficient and the constant term.
Rational Zero Test
If a polynomial f (x)  an xn  an1xn1 
 a1x  a0 has
integer coefficients, then any rational zero of f ( x)
has the form
p
q
where p is a factor of the constant term, q is a factor
of the lead coefficient, and p and q have no common
factors.
Example: Given f (x)  x4  x3  x2 3x  6
a. List all possible rational zeros of f
b. Use synthetic division to determine the
rational zeros of f.
c. Use your result to write the factors of f over
the real numbers.
Example: Given f (x)  2x3  3x2 8x  3
a. List all possible rational zeros of f
b. Use synthetic division to determine the
rational zeros of f.
c. Use your result to write the factors of f over
the real numbers.