The Fundamental Theorem of Algebra, Zeros and Their Multiplicities:
We have already seen that an nth degree polynomial can have at most n real
zeros. In the complex number system an nth degree polynomial has exactly n
zeros, and so can be factored into exactly n linear factors. This fact is a
consequence of the Fundamental Theorem of Algebra which was proved by
German mathematician C.F. Gauss in 1799.
The Fundamental Theorem of Algebra and Complete Factorization:
Every polynomial P (x)  a n x n  a n 1 x n 1        a1 x  a0 (n  1, a n  0) with
complex coefficients has at least one complex zero.
(Since every real number is also a complex number this theorem also applies to
polynomials with real coefficients)
Complete Factorization Theorem:
If P (x ) is a polynomial of degree n  1, then there exists complex numbers
a, c1 , c2 ,..........cn (with a  0 ) such that P (x)  a( x  c1 )( x  c2 )    ( x  cn )
Remember: When finding the zeros of a polynomial function first you must factor
the polynomial into linear and or non factorable quadratic factors. To find the
zeros of the non factorable quadratic, use completing the square or the quadratic
formula.
Factor the following completely:
1)
Let P( x)  x 3  x 2  81x  81
a) Find all the zeros
b) Find the complete factorization of P
2)
Let P( x)  x 3  3x 2  x  3
a) Find all the zeros
b) Find the complete factorization of P
3)
Let P( x)  x 3  2 x  4
a) Find all the zeros
4)
b) Find the complete factorization of P
Let P( x)  x 4  3x 3  7 x 2  21x  26
a) Find all the zeros
b) Find the complete factorization of P
Zeros and Their Multiplicities:
In the Complete Factorization Theorem the numbers c1 , c2 ,..........cn are the zeros
of P. These zeros need not all be different. If the factor ( x  c ) appears k times in
the complete factorization of P (x ) . Then we say that c is a zero of multiplicity k
Example:
P( x)  ( x  1) 3 ( x  2) 2 ( x  3) 5
This polynomial has zeros of:
1 (multiplicity 3)
-2 (multiplicity 2)
-3 (multiplicity 5)
Zeros Theorem:
Every polynomial of degree n  1 has exactly n zeros, provided that a
zero of multiplicity k is counted k times.
Degree
1
2
Polynomial
P( x)  x  4
P( x)  x 2  10 x  25
 ( x  5)( x  5)
Number of Zeros
1
2
3
P( x)  x 3  x
 x( x  i )( x  i )
0, i,  i
3
4
P( x)  x 4  18 x 2  81
3i (multiplicity 2)
 3i (multiplicity 2)
4
0 (multiplicity 3)
1 (multiplicity 2)
5
 ( x  3i) 2 ( x  3i) 2
5
P( x )  x 5  2 x 4  x 3
 x 3 ( x  1) 2
5)
Zero(s)
4
5 (multiplicity 2)
Find the complete factorization and all five zeros of the polynomial
P( x)  3x 5  24 x 3  48 x .