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4.6 Complex Zeros; Fundamental Theorem of Algebra
***THE FUNDAMENTAL THEOREM OF ALGEBRA
In the complex numbering system consisting of all real and imaginary numbers, if
f(x) is a polynomial of degree n (n>0) with complex coefficients, then the equation
f(x)=0 has exactly n roots.
***COMPLEX CONJUGATES THEOREM
If f(x) is a polynomial with real coefficients, and a+bi is an imaginary root of the
equation f(x)=0 then a-bi is also a root.
***Suppose f(x) is a polynomial with rational coefficients, and a and b are rational
numbers such that b is irrational. If a + b is a root of the equation f(x) =0 ,
then a - b is also a root.
***For the equation f ( x )  a n x n  a n 1 x n 1  ...  a 0 , where a 0  0
The sum of the roots is
the product of the roots is
 a n 1
an
 a0
if n is even
a
n


 a
 0 if n is odd
 a n
***Quadratic:
x2  sumx  product  0
***Every complex polynomial function f (x) of degree n  1 can be factored into
n linear factors of the form f ( x)  an ( x  r1 )( x  r2 )  ...  ( x  rn ) where an , r1 , r2 ,..., rn
are complex numbers.
Examples:
1. If 1+i is a root of P(x) with real coefficients, then name the other root.
2. If –3-4i is a root of P(x) with real coefficients, then name the other root.
3
 5 is a root of the equation 4 x 3  16 x 2  x  11  0 , then name the other
2
root.
3. If
4. If the degree is 4, find the remaining zeros if : 2  i,  i
x2  sumx  product  0
5. Form a polynomial f(x) with real coefficients having the given degree and
zeros. Degree 3; zeros 4, 1  2i
x2  sumx  product  0
6. Use the zero to find the remaining zeros: f (x)  x 3  3x 2  25x  75 ; zero : 5i
x2  sumx  product  0
7. Find the complex zeros: f (x)  x 3  35x 2  113x  65